--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)
--At the time of Athan and between it and the Iqamah:
Anas (RA) narrated that Allah’s Messenger (SAW) 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)]
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] ”
--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).
--While Making Sujoodrostrating
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 #1175.”
--When it is raining
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 #2540, ibn Majah
--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 #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.
--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]
--In the middle of the night
Abu Hurayrah (radhi allahu anhu) reported that the Messenger of Allah (sallallahu alaihi wa-sallam) said: “Our Lord descends every night to the lowest heaven, when only one third of the night has remained. He says: “Who will invoke Me, so that I may give him? Who will seek My forgiveness, so that I may forgive him.” [Saheeh al-Bukharee and Saheeh Muslim] In another report, he adds: “Then Allah extends His Hand and says: “Who wants to invest (good deeds) with the One who is not wasteful or unjust?” He continues to say this until the dawn arrives.”