It is recommended for a Muslim to take advantage of the closeness of Allah while making sujud (prostration) to make dua`a’ (supplication). 

The Prophet (peace be upon him) recommended making supplication during prostration as it is the best time of supplication, Abu Hurairah (may Allah be pleased with him) narrated that the Prophet (peace be upon him) said, “The closest that any one of you may be to his Lord is when he is prostrating, so say a lot of duaa (supplication) at that time.” (Muslim) 

It is also recommended to make supplication after the tasleem at the end of the prayer:

Abu Hurairah (may Allah be pleased with him) narrated that the Messenger of Allah (peace be upon him) said, “He who recites after every prayer: Subhan-Allah (Allah is free from imperfection) thirty-three times; Al-hamdu lillah (praise be to Allah) thirty-three times; Allahu Akbar (Allah is Greatest) thirty-three times; and completes the hundred with: La ilaha illallahu, wahdahu la sharika lahu, lahul-mulku wa lahul-hamdu, wa Huwa ‘ala kulli shai’in Qadir (there is no true god except Allah. He is One and He has no partner with Him. His is the sovereignty and His is the praise, and He is Omnipotent), will have all his sins pardoned even if they may be as large as the foam on the surface of the sea.” (Muslim) 

To know more about the ruling of prostration during prayer, watch this interesting video with Dr. Muhammad Salah. 

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Source: Huda Youtube Channel.