Acceptance of Dua
There are certain times du'a (supplication) is more likely to be accepted by Allah Subhanahu wa Ta'ala as mentioned by the Most Beloved to Allah Salla Allahu 'alayhi wa 'aalihi wa Sallam. These times are as follows:
The Last Third Of The Night
Hadrat Abu Hurairah Radi Allahu anhu narrated that Allah's Messenger Salla Allahu 'alayhi wa 'aalihi wa Sallam said: 'In the last third of every night our Rabb (Cherisher and Sustainer) (Allah Subhanahu wa ta'ala) 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]
Hadrat Amr ibn Absah narrated that the Most Beloved Prophet Salla Allahu 'alayhi wa 'aalihi wa Sallam 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]
Late at night
When people are sleeping and busy with worldly pleasures Allah Subhanahu wa ta'ala gives the believers an opportunity, or an answer hour if they can fight sleep and invoke Allah Subhanahu wa ta'ala for whatever they need. The Prophet Salla Allahu 'alayhi wa 'aalihi wa Sallam 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 #757]
Between Adhan and Iqamah
Hadrat Anas Radi Allahu anhu narrated that Allah's Beloved Messenger Salla Allahu 'alayhi wa 'aalihi wa Sallam said: 'A supplication made between the Adhan and Iqama is not rejected.'
[Ahmad, abu Dawud #521, at-Tirmidhi #212, Sahih al-Jami #3408, an-Nasai and Ibn Hibban graded it sahih (sound)]
An Hour On Friday
Narrated Abu Hurairah Radi Allahu anhu : Allah's Messenger Salla Allahu 'alayhi wa 'aalihi wa Sallam 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 Subhanahu wa ta'ala, then Allah Subhanahu wa ta'ala will definitely meet his demand.' And he (the Prophet Salla Allahu 'alayhi wa 'aalihi wa Sallam 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 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).
While Drinking Zam zam Water
Hadrat Jaber Radi Allahu anhu narrated that Allah's Messenger Salla Allahu 'alayhi wa 'aalihi wa Sallam said: 'Zamzam water is for what it is drunk for.' [Ahmad 3: 357 and Ibn Majah #3062]. This means that when you drink Zamzam water you may ask Allah Subhanahu wa ta'ala for anything you like to gain or benefit from this water such as healing from illness.... etc.
While Prostrating
Hadrat Abu Hurairah Radi Allahu anhu narrated that Allah's Messenger Salla Allahu 'alayhi wa 'aalihi wa Sallam, said: 'The nearest a slave can be to his Lord is when he is prostrating, so invoke (supplicate) Allah Subhanahu wa ta'ala much in it. [Muslim, abu Dawud, an-Nasa'i and others, Sahih al-Jami #1175]
When a Muslim is in his Salat (prayer) he is facing Allah Subhanahu wa ta'ala and when he prostrates he is the nearest he can be to Allah Subhanahu wa ta'ala so it is best to invoke Allah Subhanahu wa ta'ala 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.
When Waking Up at Night
Narrated Ubada Bin As-Samit that Allah's Messenger Salla Allahu 'alayhi wa 'aalihi wa Sallam 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 Subhanahu wa ta'ala He is the only one who has no partners. His is the kingdom and all the praises are for Allah Subhanahu wa ta'ala All the glories are for Allah Subhanahu wa ta'ala And none has the right to be worshipped but Allah Subhanahu wa ta'ala and Allah Subhanahu wa ta'ala is the most Great and there is neither might nor power except with Allah Subhanahu wa ta'ala and then says, Allahumma ighfir li (O Allah! Forgive me) or invokes Allah Subhanahu wa ta'ala, he will be responded to and if he makes ablution and performs Salat (prayer), his Salat (prayer) will be accepted. [Sahih al-Bukhari]
At The End Of The Obligatory Salat:
Narrated Abu Omamah Radi Allahu anhu : that Allah's Beloved Messenger Salla Allahu 'alayhi wa 'aalihi wa Sallam was asked, O Messenger of Allah, which supplication is heard (by Allah Subhanahu wa ta'ala, 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).
The Night Of 'Qadr' (Decree)
This night is the greatest night of the year. This is the night which the almighty Allah Subhanahu wa ta'ala 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.
During The Rain
Narrated Sahel Ibn Sa'ad Radi Allahu anhu: that the Most Beloved Messenger of Allah Salla Allahu 'alayhi wa 'aalihi wa Sallam 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 #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 #1469]. The time of the rain is a time of mercy from Allah Subhanahu wa ta'ala so, one should take advantage of this time when Allah Subhanahu wa ta'ala is having mercy on His slaves.
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 #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 #2106, al-Sahihah #1413]
The Beloved Messenger of Allah Salla Allahu 'alayhi wa 'aalihi wa Sallam 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 Subhanahu wa ta'ala' [Sahih Al-Bukhari and Muslim]
The Beloved Messenger Salla Allahu 'alayhi wa 'aalihi wa Sallam declared, 'Three men whose du'a 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 Beloved Messenger Salla Allahu 'alayhi wa 'aalihi wa Sallam 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 Subhanahu wa ta'ala when he invokes Allah Subhanahu wa ta'ala to retain his rights from the unjust one or oppressor. Allah Subhanahu wa ta'ala has sworn to help the one who is suffering from injustice sooner or later as the Messenger of Allah Salla Allahu 'alayhi wa 'aalihi wa Sallam said.
The Traveller
The Most Beloved of Allah Salla Allahu 'alayhi wa 'aalihi wa Sallam said; Three supplications will not be rejected (by Allah Subhanahu wa ta'ala), 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 Subhanahu wa ta'ala 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.
The Beloved Messenger of Allah Salla Allahu 'alayhi wa 'aalihi wa Sallam said; Three supplications will not be rejected (by Allah Subhanahu wa ta'ala), 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]
Du'a after praising Allah and giving salat on the Beloved Prophet Salla Allahu 'alayhi wa 'aalihi wa Sallam in the tashahhud at the end of salat.
Narrated Faddalah ibn Ubayd Radi Allahu anhu: that the Messenger of Allah Salla Allahu 'alayhi wa 'aalihi wa Sallam 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 Salla Allahu 'alayhi wa 'aalihi wa Sallam, then let him pray for whatever he wants' [abu Dawud #1481, at-Tirmidhi #3477]
In another hadith; Baqiy ibn Mukhallid Radi Allahu anhu narrated that that the Beloved Messenger of Allah Salla Allahu 'alayhi wa 'aalihi wa Sallam said: 'Every du'a is not responded to until one sends blessings upon the Prophet Salla Allahu 'alayhi wa 'aalihi wa Sallam' [al-Bayhaqi]
In another hadith; Hadrat Umar Radi Allahu ta'ala anhu narrated that the Most Beloved Prophet Salla Allahu 'alayhi wa 'aalihi wa Sallam 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 Beloved Prophet Salla Allahu 'alayhi wa 'aalihi wa Sallam' [at-Tirmidhi #486]