Answered by Shaykh Amjad Rasheed
Is it permissible for a muhdith (one who is not on wudu) to touch a mushaf (copy of the Quran) or to carry it when it is in its own case made of carton? Is it permissible for him or for the a person with wudu to carry a small mushaf in the pocket of his shirt or in his pant pocket?
It is not permissible for the muhdith to carry a mushaf in a case, unless there is something else in the case with it and he doesn’t intend to carry the mushaf alone, then it is permissible. It is permissible to put a mushaf in the pocket of one’s shirt because in it, it is usually protected from belittlement. As for the pant pocket, if you mean the back pocket, it is not permissible because of the belittlement that is involved, especially because sitting on it is a very major haram. If it is in the front, then it is protected from belittlement, so it is permissible, if not though, then it is not [s. permissible]. Also, it is a part of good manners that one not describe the mushaf as small, rather it should be said, “of a small print, for example.”
– Amjad Rasheed
(Translated by Shazia Ahmad)