The Prophet would fast most of the month of Sha’ban. ‘Aishah said: “I never saw the Messenger of Allah fast a complete month save for Ramadan, and I have never seen him fast more in a month than he did in Sha’ban.” This is related by al-Bukhari and Muslim.
Usamah ibn Zaid inquired: “O Messenger of Allah, I never find you fasting in any month like you do during the month of Sha’ban.” The Prophet responded: “That is a month the people neglect. It comes between Rajab and Ramadan. It is a month in which the deeds are raised to the Lord of the Worlds. I love that my deeds be raised while I am fasting.” This is related by Abu Dawud, an-Nasa’i, and by Ibn Khuzaimah in his Sahih.
Some people fast on the 15th of Sha’ban in particular, thinking that that day contains more virtues than the other days. This is an unsubstantiated claim.