1) When rain falls, as it said in the hadeeth of Sahl ibn Sa’d that is attributed to the Prophet (peace and blessings of Allaah be upon him): “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

2)Between the adhaan and the iqaamah. It is narrated in a saheeh report that the Prophet (peace and blessings of Allaah be upon him) said: “A du’aa’ offered between the adhaan and iqaamah is not rejected.” Narrated by Abu Dawood (521) and al-Tirmidhi (212).

3) When drinking Zamzam water. It was narrated from Jaabir (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said: “Zamzam water is for that for which it is drunk.” Narrated by Ahmad

4&5) It was narrated in a saheeh report that our Prophet (peace and blessings of Allaah be upon him) said: “There are three prayers that are not rejected: the prayer of a father for his child, the prayer of the fasting person and the prayer of the traveller.” Narrated by al-Bayhaqi

6) At a certain time of the night, as the Prophet (peace and blessings of Allaah be upon him) said: “During the night there is a time when the Muslim does not ask for the good of this world and the Hereafter but it will be given to him, and that happens every night.” Narrated by Muslim, 757.

7)A certain time on Friday.  The Messenger of Allaah (peace and blessings of Allaah be upon him) mentioned Friday and said: “During it there is a time when a Muslim slave does not stand and pray and ask Allaah for something, but He will give it to him,” and he gestured with his hand to indicate how short that time is. Narrated by al-Bukhaari, 935; Muslim, 852. (most probably between asr and magrib)

8)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, 482