There is no evidence of the Prophet (Peace be upon him) worshipping Allah in any distinctive way on Laylat ul-Qadr (the Night of Decree, which falls in the final odd nights of Ramadan). However, when the last ten days began, the Prophet (peace be upon him) would awaken himself and his family and they would devote themselves to abundant worship. Therefore, along with fulfilling the faraiz, every Muslim should engage in nafil prayers, recitation of the Quran, sending durood and salam upon the Prophet, duas, and zikr during the last 10 nights of Ramadan, so that they may attain the rewards of various a’maal.
Furthermore, Hazrat Aisha (may Allah be pleased with her) asked the Prophet (peace be upon him) regarding this matter: "O Messenger of Allah! If I know which night is Laylat al-Qadr, what du'a should I make?" The Prophet (peace be upon him) said:
اللهم إنك عفو تحب العفو فاعف عنى
"O Allah! You are indeed the Most Forgiving, and You love forgiveness, so forgive me." (Jami' ut-Tirmidhi, Hadith No. 3513)
Therefore, one should recite this du'a abundantly during these nights.