In Sha'ban are we allowed to fast Monday and Thursday up until the beginning of Ramadan or is it Sunnah to stop half of Sha'ban?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

Fasting on Mondays and Thursdays is permissible in the last half of the month of Sha'baan if a person is used to regularly fast on these days. The Prophet said: "None of you should fast a day or two before the month of Ramadan unless he has the habit of regularly fasting on that day, in which case he can fast that day." [Al-Bukhari and Muslim]

However, there is no contradiction between this Hadeeth and the Hadeeth reported by At-Tirmithi which reads: “Don't fast after the fifteenth of Sha'baan.”

The scholars reconciled between these two Ahadeeth that the one who fasted in the first half of Sha'baan can also fast in the second half of Sha'baan. They said that the prohibition applies to those who did not fast in the first half of Sha'baan. The evidence for this is that the Prophet used to fast most of the month of Sha'baan. [Al-Bukhari, Ahmad and An-Nasaa'i]

No doubt that most of Sha'baan includes a portion from the second half of this month.

Allaah Knows best.