Is it permissible to place one's hand on the Mus'haf and swear by the truth during menstruation? And does that entail a sin requiring expiation?

The questioning sister must refrain from touching the Qur’an during menstruation. This is the view of the majority of scholars and the imams of the four madhhabs. She must repent to Allah Almighty for this forbidden act, and there is no expiation for her other than repentance.