The holy Prophet Muhammad (صلى الله عليه وسلم) used to keep 6 fasts of Shawwal after Ramadan. Also he used to keep fast in most of the days of Shaban, Yaum Ashura, 13, 14, 15 of each month and in the starting days of Zul Hijjah. [Mishkah p. 178]
The holy Prophet Muhammad (صلى الله عليه وسلم) used to pray the whole night during the night of Arafah, the night of Tarvia, the night of Eid al-Fitr, the night of Bara’ah, the night of Qadr, the night of Eid al-Azha as well the nights of last ten days of Ramadan. Also he instructed the Ummah to pray in these nights. Instead of these particular nights, the holy Prophet Muhammad (صلى الله عليه وسلم) offered tahajjud regularly each night.
Allah (Subhana Wa Ta’ala) knows Best
Darul Uloom Deoband