Concerning the sighting of the crescent moon
ʿAbd Allah ibn ʿUmar (may Allah be pleased with him) said: “Whenever the Messenger of Allah ﷺ saw the new moon he would say, ‘Allahu akbar. O Allah, let it rise over us with security and faith, with safety and Islam, and with success in what You love and approve. Our Lord and yours is Allah.’” Reported by al-Darimi; al-Tirmidhi transmitted a shorter version through the narration of Talhah.

On fasting and breaking the fast
Abu Hurayrah (may Allah be pleased with him) reported that the Messenger of Allah ﷺ said, “Three supplications are never rejected: the supplication of a fasting person until he breaks his fast, the supplication of a just ruler, and the supplication of one who has been wronged.” Al-Tirmidhi graded it as a ḥasan (sound) hadith.

Ibn Abi Mulaykah narrated that ʿAbd Allah ibn ʿAmr (may Allah be pleased with them both) said, “I heard the Messenger of Allah ﷺ say, ‘When the fasting person breaks his fast, he has a supplication that will not be rejected.’” Ibn Abi Mulaykah added, “Whenever ʿAbd Allah ibn ʿAmr broke his fast, I heard him say: ‘O Allah, I ask You…’”