1. The Last Third Of The Night
Abu Hurairah (RA) narrated that Allah’s Messenger (SAW) said: 'In the last third of every night our Rabb (Cherisher and Sustainer) (Allah (SWT)) descends to the lowermost heaven and says; "Who is calling Me, so that I may answer him? Who is asking Me so that may I grant him? Who is seeking forgiveness from Me so that I may forgive him?."' [Sahih al-Bukhari, Hadith Qudsi]

Amr ibn Absah narrated that the Prophet said: 'The closest any worshipper can be to His Lord is during the last part of the night, so if you can be amongst those who remember Allah at that time, then do so.'[at-Tirmidhi, an-Nasa'i, al-Hakim - Sahih]
 

2. Late at night
When people are sleeping and busy with worldly pleasures Allah (SWT) gives the believers an opportunity, or an answer hour if they can fight sleep and invoke Allah (SWT) for whatever they need. The Prophet (SAW) said: 'There is at night an hour, no Muslim happens to be asking Allah any matter of this world or the Hereafter, except that he will be given it, and this (occurs) every night.'
[Muslim No. 757]

3. Between Adhan and Iqamah
Anas (RA) narrated that Allah’s Messenger (SAW) said: 'A supplication made between the Adhan and Iqama is not rejected.'
[Ahmad, abu Dawud No. 521, at-Tirmidhi No. 212, Sahih al-Jami No. 3408, an-Nasai and Ibn Hibban graded it sahih (sound)]
 

4. An Hour On Friday
Narrated Abu Hurairah (RA): Allah’s Messenger (SAW) talked about Friday and said: 'There is an hour on Friday and if a Muslim gets it while offering Salat (prayer) and asks something from Allah (SWT), then Allah (SWT) will definitely meet his demand.' And he (the Prophet (SAW) pointed out the shortness of that particular time with his hands.[Sahih al-Bukhari]

Some have said that this hour is from the time the Imam (prayer’s leader) enters the mosque on Friday’s prayer until the prayer is over (ie between the two khutbahs), whereas others have said that it is the last hour of the day (ie after the Asr prayer until the Maghrib prayer).
 

5. While Drinking Zamzam Water
Jaber (RA) narrated that Allah’s Messenger (SAW) said: 'Zamzam water is for what it is drunk for.' [Ahmad 3: 357 and Ibn Majah No. 3062]. This means that when you drink Zamzam water you may ask Allah (SWT) for anything you like to gain or benefit from this water such as healing from illness.... etc.
 

6. While Prostrating
Abu Hurairah (RA) narrated that Allah’s Messenger (SAW), said: 'The nearest a slave can be to his Lord is when he is prostrating, so invoke (supplicate) Allah (SWT) much in it. [Muslim, abu Dawud, an-Nasa'i and others, Sahih al-Jami No. 1175]

When a Muslim is in his Salat (prayer) he is facing Allah (SWT) and when he prostrates he is the nearest he can be to Allah (SWT) so it is best to invoke Allah (SWT) at this time. It is said that while in prostration, one should not ask for worldy needs (ie a nice car, a new job, etc), but for the Hereafter.
 

7. When Waking Up at Night
Narrated Ubada Bin As-Samit that Allah’s Messenger (SAW) said: 'Whomever wakes up at night and says La ilaha illallahu wahdahu la shrika lahu lahulmulku, wa lahul hamdu, wa huwa ala kulli shai'in qadir. Alhamdu lillahi, wa subhanallahi wa la ilaha illallahu, wallah akbir, wa la hawla wala quwata illa billah (none has the right to be worshipped but Allah (SWT) He is the only one who has no partners. His is the kingdom and all the praises are for Allah (SWT) All the glories are for Allah (SWT) And none has the right to be worshipped but Allah (SWT) and Allah (SWT) is the most Great and there is neither might nor power except with Allah (SWT) and then says, Allahumma ighfir li (O Allah! Forgive me) or invokes Allah (SWT), he will be responded to and if he makes ablution and performs Salat (prayer), his Salat (prayer) will be accepted. [Sahih al-Bukhari]
 

8. At The End Of The Obligatory Salat:
Narrated Abu Omamah (RA): that Allah’s Messenger (SAW) was asked, O Messenger of Allah, which supplication is heard (by Allah (SWT), he said the end of the night and at the end of the obligatory Salat (prayer) [at-Tirmidhi]. This time is after saying 'At-tahyat' , and before making Tasleem (finishing prayer)
 

9. The Night Of 'Qadr' (Decree)
This night is the greatest night of the year. This is the night which the almighty Allah (SWT) said about it, "The night of Al-Qadar (Decree) is better than a thousand months." [Surah al-Qadr, 97: 3]

The Night of Decree is one of the odd nights of the last ten nights of the blessed month of Ramadan. The angels descend down to the earth, and the earth is overwhelmed with peace and serenity until the break of dawn and when he doors of Paradise are opened, the worshipper is encouraged to turn to Allah to ask for his needs for this world and the Hereafter.
 

10. During The Rain
Narrated Sahel Ibn Sa'ad (RA): that the Messenger of Allah (SAW) said: 'Two will not be rejected, Supplication when the Adhan (call of prayer) is being called, and at the time of the rain'. [Al-Hakim 2: 114, and Abu Dawud No. 2540, ibn Majah]

'Seek the response to your du'as when the armies meet, and the prayer is called, and when rain falls' [reported by Imam al-Shafi' in al-Umm, al-Sahihah No. 1469]. The time of the rain is a time of mercy from Allah (SWT) so, one should take advantage of this time when Allah (SWT) is having mercy on His slaves.
 

11. At the Adhan
'Seek the response to your du'as when the armies meet, and the prayer is called, and when rain falls' [reported by Imam al-Shafi' in al-Umm, al-Sahihah No. 1469]

In another hadith; 'When the prayer is called, the doors of the skies are opened, and the du'a is answered' [al-Tayalisi in his Musnad No. 2106, al-Sahihah No. 1413]
 

12. The One Who Is Suffering Injustice and Opression
The Messenger of Allah (SAW) said to Mua'ad Ibn Jabal (RA), 'Beware of the supplication of the unjustly treated, because there is no shelter or veil between it (the supplication of the one who is suffering injustice) and Allah (SWT)' [Sahih Al-Bukhari and Muslim]

The prophet (SAW) declared, 'Three men whose dua is never rejected (by Allah) are: the fasting person until he breaks his fast (in another narration, when he breaks fast), the just ruler and the one who is oppressed.'[Ahmad, at-Tirmidhi - Hasan]

In another hadith; The Prophet (SAW) declared: 'Three du'as are surely answered: The du'a of the oppressed, the du'a of the traveler, and the du'a of the father/mother (upon their child)'. The One who is suffering injustice is heard by Allah (SWT) when he invokes Allah (SWT) to retain his rights from the unjust one or oppressor. Allah (SWT) has sworn to help the one who is suffering from injustice sooner or later as the Messenger of Allah (SAW) said.
 

13. The Traveler
The Messenger of Allah (SAW) said; Three supplications will not be rejected (by Allah (SWT)), the supplication of the parent for his child, the supplication of the one who is fasting, and the supplication of the traveler. [al-Bayhaqi, at-Tirmidhi - Sahih]

During travel supplication is heard by Allah (SWT) if the trip is for a good reason, but if the trip is for a bad intention or to perform illegal things (making sins) this will not apply to it.
 

14. The Parent's Supplication for their Child
The Messenger of Allah (SAW) said; Three supplications will not be rejected (by Allah (SWT)), the supplication of the parent for his child, the supplication of the one who is fasting, and the supplication of the traveler. [al-Bayhaqi, at-Tirmidhi - Sahih]
 

15. Dua after praising Allah and giving salat on the Prophet (SAW) in the tashahhud at the end of salat.
Narrated Faddalah ibn Ubayd (RA): that the Messenger of Allah (SAW) said: 'When anyone of you makes du'a, let him start by glorifying his Lord and praising Him, then let him send blessings upon the Prophet (SAW), then let him pray for whatever he wants'
[abu Dawud No. 1481, at-Tirmidhi No. 3477]

In another hadith; Baqiy ibn Mukhallid (RA) narrated that that the Messenger of Allah (SAW) said: 'Every du'a is not responded to until one sends blessings upon the Prophet (SAW)' [al-Bayhaqi]

In another hadith; Umar (RA) narrated that the Prophet (SAW) said: 'Du'a is detained between the heavens and the earth and no part of it is taken up until you send blessings upon your Prophet (SAW)' [at-Tirmidhi No. 486]

After a person has finished his tashahhud and before saying the 'salam', supplication at this time is one likely to be responded to.
Ibn Mas'ud narrates: I was once praying, and the Prophet (SAW), Abu Bakr and Umar (were all present). When I sat down (in the final tashahhud), I praised Allah, then sent salams on the Prophet, then started praying for myself. At this, the Prophet (SAW) said:
'Ask, and you shall be given it! Ask, and you shall be given it!' [at-Tirmidhi No. 593 - hasan, Mishkat al-Misbah No. 931]
 

16. The dua of a Muslim for his absent brother or sister Muslim stemming from the heart.
The prophet (SAW) said: 'There is no believing servant who supplicates for his brother in his absence where the angels do not say, 'the same be for you'' [Muslim]
 

17. Dua on the Day of Arafat
The Messenger of Allah (SAW) said: 'The best supplication is the supplication on the day of Arafat'.  [at-Tirmidhi and Malik]

The day of Arafat is the essence and pinnacle of Hajj. On this great and momentous day, millions of worshippers gather together on one plain, from every corner of the world, with only one purpose in mind - to respone to the call of their Creator. During this auspicious day, Allah does not refuse the requests of His worshippers.
 

18. Dua during the month of Ramadan
Ramadan is month full of many blessings, thus the du'a of Ramadan is a blessed one. This can be inferred from the Prophet (SAW) saying: 'When Ramadan comes, the Doors of Mercy (another narration says Paradise) are opened, and the doors of Hell are closed, and the Shaitans are locked up' Thus, it is clear that du'a during Ramadan has a greater chance of being accepted, as the Gates of Paradise and Mercy are opened. [Sahih al-Bukhari No. 1899, Muslim No. 1079 and others]
 

19. Dua when the Armies meet
When the Muslim is facing the enemy in battle, at this critical period, the du'a of a worshipper is accepted.

Sahl ibn Sa'd (RA) narrtaed the the Prophet (SAW) said: 'Two duas are never rejected, or rarely rejected: the du'a during the call for prayer, and the du'a during the clamity when the two armies attack each other' [Abu Dawud No. 2540, ibn Majah, al-Hakim]

20. When Muslims gather for the purpose of invoking and remembering Allah (Dhikrullah).
The Prophet (SAW) said: If a group of people sit together remembering Allah, the angels will circle them, mercy will shroud them, peace will descend onto them and Allah will remember them among those with Him.  [Muslim]
 

21. First Ten days of Dhul-Hijjah
The Prophet (SAW) said: 'There are no days during which good deeds are more beloved to Allah than during these ten days'
[Sahih al-Bukhai No. 969 and others]
 

22. Dua when the heart reaches out to Allah and is ready to be totally sincere
 

23. At Midnight
Abu Umamah (RA) said, the Prophet (SAW) was questioned; 'Which du'a is heard (by Allah)?' He answered, 'At midnight and at the end of every obligatory prayer.' [at-Tirmidhi - Hasan]
 

25. Dua of people after the death of a person
In a long hadith, Umm Salamah (RA) narrated that the Prophet (SAW) said, when Abu Salamah had just passed away, and had closed his eyes, 'Do not ask for yourselves anything but good, for the angels will say 'Ameen' to all that you ask for. O Allah, forgive Abu Salamah, and raise his ranks among those who are guided.' [Muslim, abu Dawud, Ahmad]
 

26. Dua of the one fasting until he breaks his fast.
The Messenger of Allah (SAW) said; Three supplications will not be rejected (by Allah (SWT)), the supplication of the parent for his child, the supplication of the one who is fasting, and the supplication of the traveler. [al-Bayhaqi, at-Tirmidhi - Sahih]
 

27. Dua of the one fasting at the time of breaking fast

The prophet (SAW) declared, 'Three men whose dua is never rejected (by Allah) are: when a fasting person breaks fast (in another narration, the fasting person until he breaks his fast), the just ruler and the one who is oppressed.'[Ahmad, at-Tirmidhi - Hasan]
 

29. Dua of a just Ruler
The prophet (SAW) declared, 'Three men whose dua is never rejected (by Allah) are: the fasting person until he breaks his fast (in another narration, when he breaks fast), the just ruler and the one who is oppressed.'[Ahmad, at-Tirmidhi - Hasan]
 

30. Dua of a son or daughter obedient to his or her parents
It is well known in the story narrated in hadith os three men who were trapped by a huge stone in a cave. One of them who was kind to kis parents asked Allah to remove the stone, and his du'a was answered. [Sahih al-Bukhari 3:36 No. 472]
 

31. Dua immediately after wudu`
Umar ibn Al-Khattab reported that the Prophet (SAW) said: 'There is not one of you that makes wudu, and does it perfectly, then says: I testify that there is no diety worthy of worship except Allah. he is Alone, having no partners. And I bear witness that Muhammad is His slave and messenger', except that the eight gates of Paradise are opened for him, and he can enter into it through whichever one he pleases' [Muslim No. 234, abu Dawud No. 162, Ahmad, an-Nasa'i] Saying what has been mentioned in the hadith that is directly related to it (ablution)
 

32. Dua after stoning the Jamarat at Hajj
The stoning of the small Shaitan (jamrat sugra), or the middle Shaitan (jamrat wusta) pillars during Hajj. It is narrated that the Prophet (SAW) would stone the small Jamarah (one of the three pillars that is stoned in the last days of Hajj), then face the qiblah, raise his hands, and make du'a for a long time. He would then stone the middle Jamarah and do the same. When he stoned the large Jamarah, he would depart without making any du'a. [Sahih al-Bukhari No. 1753 and others]
 

33. At the Crowing of a Rooster
Abu Hurairah (RA) narrated that the Prophet (SAW) said: 'When you hear a rooster crowing, then ask Allah for His bounties, for it has seen an angel, and when you hear a donkey braying, then seek refuge in Allah from Shaitan, for it has seen a Shaitan' [Sahih al-Bukhari, Muslim, Ahmad, Sahih al-Jami No. 611]
 

34. Du'a made inside the Ka'bah
The Ka'bah is a sanctuary that has no comparison in the entire world. The du'a of one who prays inside the hijr is considered as being made inside the ka'bah, as it is part of the house (Baitullah). [This is the semicircle to the right of the Ka'bah if you face the door, opposite to the Yemeni corner and the Black stone wall.]

Usamah ibn Zayd narrated, 'When the Prophet (SAW) entered the House (Ka'bah), he made du'a in all of its corners [Muslim 2: 968 and others]
 

35. Du'a on the mount of Safa or Marwah during Umrah or Hajj
It is narrated that the Prophet (SAW) would make long du'as at Safa and Marwah. [Muslim No. 1218 and others]
 

36. Dua at any of the holy sites.
 

37. While reciting Surah al-Fatihah
The Prophet (SAW) said that Allah the Exalted had said: ' I have divided the prayer into two halves between Me and My servant, and My servant will receive what he asks for. When the servant says: Praise be to Allah, the Lord of the universe, Allah the Most High says: My servant has praised Me. And when he (the servant) says: The Most Compassionate, the Merciful, Allah the Most High says: My servant has lauded Me. When he (the servant) says: Master of the Day of Judgment, He remarks: My servant has glorified Me, and sometimes He will say: My servant entrusted (his affairs) to Me. When he (the worshipper) says: Thee do we worship and of Thee do we ask help, He (Allah) says: This is between Me and My servant, and My servant will receive what he asks for. Then, when he (the worshipper) says: Guide us to the straight path, the path of those to whom Thou hast been Gracious -- not of those who have incurred Thy displeasure, nor of those who have gone astray, He (Allah) says: This is for My servant, and My servant will receive what he asks for. [Muslim 4: 395]
 

38. Saying 'Ameen' during prayer
After finishing the recitation of al-Fatihah, the saying of'Ameen' in congregation. The Prophet (SAW) said: 'When the Imam says 'Ameen', then recite it behind him (as well), because whoever's Ameen coincides with the Ameen of the angels will have all of his sins forgiven.' [Sahih al-Bukhari No. 780, Muslim No. 410 and others]
 

39. While visiting the sick, and dua made by the sick
Umm Salamah narrated that the Prophet (SAW) said: 'When you visit the sick, or the dead then say good, because the angels say 'Ameen' to whatever you say [Muslim No. 2126]

Ali (RA) reported that the Prophet (SAW) said: 'When a Muslim visits his sick Muslim brother in the morning, seventy thousand angels make dua for his forgiveness till the evening. And when he visits him in the evening, seventy thousand angels make dua for his forgiveness till the morning, and he will be granted a garden for it in Jannah.'
[at-Tirmidhi, abu Dawud