The Best times are:
Before Fajr (dawn),
Narrated Ubada Bin As-Samit (Radhi Allahu Anhu) that Allah's Messenger (Sallallahu 'Alayhi wa Sallam) said: “ Whomever wakes up at night and says La ilaha illallahu wahdahu la shrika lahu lahulmulku, walahul hamdu, wa huwa ala kulli shai'in qadir. Subhanallahi wa Alhamdulillah wa La ilaha illallahu, wallah akbir, wa la hawla wala quwata illa billah ‘Aliyyul-Adheem. (None has the right to be worshipped but Allah Alone. Who has no partners. His is the kingdom and to Him belongs all praise, and He is Able to do all things. Glory is to Allah, Praise is to Allah, there is none worthy of worship except Allah, Allah is the greatest. There is no power or might and power except by Allah’s leave the exalted, the Mighty) 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.” (Reported by Bukhari).
The Last Third of the Night.
Abu Hurairah, (Radhi Allahu
Anhu) said: The Prophet,(Sallallahu 'Alayhi wa Sallam) said : "Almighty Allah
manifests Himself at the first heaven, during the last third of night, and says:
‘Is there any suppliant so that I answer him? Is there anyone asking so that I
fulfil his need? Is there any seeker of forgiveness so that I forgive his
sins.'”(Al-Bukhari)
At the end of the Salah (Prayers)
Narrated Abu Omamah (Radhi Allahu Anhu) that Allah's Messenger (Sallallahu 'Alayhi wa Sallam) was asked: “ O Messenger of Allah, which supplication is heard. He (Sallallahu 'Alayhi wa Sallam) said “ In the last depth (end) of the night and at the end of the obligatory Salat (prayer).” (Reported by At-Tirmidhi).
An Hour of Jumu’ah (Friday),
Narrated Abu Hurairah (Radhi Allahu Anhu), the Messenger of Allah (Sallallahu 'Alayhi wa Sallam) talked about Friday and said: “There is an hour on Friday and if a Muslim gets it while offering Salat and asks something from Allah then Allah will definitely meet his demand.” And he (Sallallahu 'Alayhi wa Sallam) pointed out the shortness of that time with his hands. (Al-Bukhari and Muslim)
From Jabir (Radhi Allahu Anhu) that the Messenger of Allah (Sallallahu 'Alayhi wa Sallam) said, "the day of Jumu’ah has twelve hours, and during one of the hours you will not find a Muslim servant (of Allah) asking Allah for something except that He will give it to him. Seek it in the last hour after the prayer of Asr" (Abu Dawud, An-Nasa’i and others.)
Although this Hadith points to the last hour of the day between Asr and Maghrib, some scholars said this hour is from the time the Imam (prayer's leader) enters the Masjid on Friday's prayer until the prayer is over , whereas others have said that it is the hour before Salatul-Jumu’ah based on other Ahadith. Allah knows best.
After Adhan
Narrated Anas (Radhi Allahu Anhu) that the Messenger of Allah (Sallallahu 'Alayhi wa Sallam) (Sallallahu 'Alayhi wa Sallam) said: “When the Adhan is proclaimed, the doors to the heaven open and the supplications are answered. (At-Tabarani)
Between the Adhan and Iqamah.
The Prophet (Sallallahu 'Alayhi wa Sallam) said:
“Dua made between Adhan and Iqamah is not rejected." The Companions said: "What shall we say, O Messenger of Allah?" "Seek Allah's grace in this world and in the afterlife", replied the Prophet (Sallallahu 'Alayhi wa Sallam) (Abu-Dawud and At-Tirmidhi)
During Prostration.
Abu Hurairah, (Radhi Allahu Anhu) said the Prophet (Sallallahu 'Alayhi wa Sallam) said: “The nearest a Muslim is to his Lord is while he is in prostration. So, make Dua in it." (Muslim, Abu Dawud and An-Nasa'i)
During Rainfall
Allah Subhanahu wa Ta'ala asks in the Glorious Qur'an:
“The tell Me about the water that you drink. Is it you who cause it from the rain clouds to come down, or are We the Causer of it to come down? If We willed, We verily could make it salty when then do you not give thanks (to Allah)” (Surah Al-Waqi’ah 56: 68-70)
Narrated Sahel Ibn Saad (Radhi Allahu Anhu) that the Messenger of Allah (Sallallahu 'Alayhi wa Sallam) said: “Two will not be rejected, Supplication after the Adhan (call of prayer) and at the time of the rain.” (Reported by Al-Hakim and Abu Dawud). This Hadith refers to both the above.
While drinking Zam Zam water
Narrated ibnJiber (Radhi Allahu Anhu) Allah's Messenger (Sallallahu 'Alayhi wa Sallam) said: “Zam Zam water is for what it is drunk for.” (Reported by Ahmed and Ibn Majah).
This means that when you drink Zam Zam water you may ask (Sallallahu 'Alayhi wa Sallam) for anything you like to gain or benefit from this water such as healing from illness.... etc.
On the day of Arafat
The Messenger of Allah (Sallallahu 'Alayhi wa Sallam) said: “ The best supplication is the supplication on the day of Arafat.” (Reported by At-Tirmidhi and Malik).
According to the majority of the scholars this Hadith applies to Pilgrims as well as non-Pilgrims.
The Night of Qadr (Decree)
This night is the greatest night of the year. Allah Subhanahu wa Ta'ala said:
“ The night of Al-Qadar (Decree) is better than a thousand month.”(Surah Al-Qadr 97: 3).
Abu Hurayrah (Radhi Allahu Anhu) narrated: "When Ramadan would come, the Messenger of Allah (Sallallahu 'Alayhi wa Sallam) would say,
“Verily, the month of Ramadan has come to you all. It is a blessed month, which Allah has obligated you all to fast. During it the gates of Paradise are opened, the gates of Hell are closed and the devils are shackled. In it there is a night that is better than one thousand months. Whoever is deprived of its good, then he has truly been deprived.'' (An-Nasa’i and Imam Ahmad)