Touching the Mushaf without Wudu
Q: Is it permissible to touch the mushaf (a copy of the Qur'an) without wudu?
A: Due to a ḥadath (major or minor), it is impermissible to touch a copy of the Qurʾān or a portion thereof without a barrier though it can be carried without touching it and its pages turned with one’s sleeve or a stick.
Whoever is required to make ghusl, including a woman during her monthly cycle, is prohibited from reciting a verse or more.
Source: Sharh al-Muntaha, Umdat al-Talib