1) WHEN STIRRING IN THE 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.
2) AT THE END OF SALAH BEFORE THE TASLEEM
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)
3) 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 - 1481, at-Tirmidhi - 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 - 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 - 593 – hasan, Mishkat al-Misbah 931]
4. 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]
5. AFTER WUDHU.
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]
(Source - www.muslimmarriageguide.com)