When Duaas are accepted?

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]

Between Adhan and Iqamah

It was narrated that Anas (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “Du’a is not rejected between the adhan and iqamah, so engage in du’a (supplication).” (Narrated by al-Tirmidhi, 212; Abu Dawood, 437; Ahmad, 12174 – this version narrated by him. Classed as saheeh by al-Albani in Saheeh Abi Dawood, 489).

Whilst fasting, Whilst travelling

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 traveller. [al-Bayhaqi, at-Tirmidhi - Sahih]

Friday

Abu Dawood (1048) and al-Nasaa’i (1389) narrated from Jaabir ibn ‘Abd-Allaah that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Friday is twelve hours in which there is no Muslim who asks Allaah for something but He will give it to him, so seek the last hour after ‘Asr.” [Classed as saheeh by al-Albaani].

2 opinions - 1) It is between the time when the imam sits down, until the prayer is over [Muslim] 2) Between asr and magrab prayer [Abu Dawood, Saheeh by al-Albaani] 2nd opinion is stronger. Opinion 1 does not mean that person busies themselves with duaa whilst the imaam is delivering the khutbah but they need to say ameen to his duaas and also they need to make duaa in their prayer, 

In prostration

The Prophet (peace and blessings of Allaah be upon him) said: “The closest that a person is to his Lord is when he is prostrating, so say a great deal of du’aa’ then.” Narrated by Muslim

Whilst praying, before the salaam

It was said: “O Messenger of Allaah, which du’aa’ is heard?” He said: “In the last third of the night, and following the prescribed prayers.” Narrated by al-Tirmidhi, classed as hasan by al-Albaani. Scholars such as Ibn Taymiyah have mentioned that it's after reciting salaam on the Prophet sws and before the salaam.

When it is raining

“There are two which will not be rejected: du’aa’ at the time of the call (to prayer) and when it is raining.” Narrated by Abu Dawood and classed as saheeh by al-Albaani in Saheeh al-Jaami.

Duaa when the battle begins

When the call for the prescribed prayers is given and when the ranks are drawn up for battle, as it says in the marfoo’ hadeeth of Sahl ibn Sa’d: “There are two that will not be rejected, or will rarely be rejected: du’aa’ at the time of the call for prayer and at the time of battle when the fighting begins.” Narrated by Abu Dawood; it is a saheeh report. See Saheeh al-Jaami’, 3079.

REF :
1) https://www.islamawareness.net/Dua/best.html 
2) https://islamqa.info/en/answers/22438/places-and-times-where-duaa-is-answered