All perfect praise be to Allah; and may His blessings and peace be upon our Prophet Mohammad and upon all his Family and Companions.
Observing fast during Sha`baan or part of it is permissible in Sharia. 'Aishah (May Allah be pleased with her) reported: “The Prophet (PBUH) did not observe voluntary Saum (fasting) so frequently during any other month as he did during Sha'ban. He observed Saum throughout the month of Sha'ban.” {Bukhari&Muslim}.
However, fasting the second half of Sha`baan, in particular, is excluded since it is considered forbidden as indicated by the following Hadeeth: Abu Hurairah (May Allah be pleased with him) reported: The Messenger of Allah (PBUH) said: "When there comes the middle of Sha'ban, don't observe Saum (fasting)."{At-Tirmithi graded it as Hasan Sahih}.
Fasting the second half of Sha`baan is permissible in the following situations:
First: The person who is used to observing fast regularly-Mondays, Thursdays and The White Days-is allowed to continue doing so during the last half of Sha`baan.
Second: The person who has started observing fast from the beginning of Sha`baan, such as fasting some days in the first half and other ones in the second half. Third: The person who has made the intention to make up for obligatory fast, vow fast, or expiatory fast. And Allah knows best.