All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Putting the Mus-haf on one’s lap or thighs when reading is not a disdain to the Quran; rather, this act is permissible and there is nothing wrong with it.
Among honoring the Mus-haf is to always look at it (i.e. to recite it) and a person should not touch it except in a state of purity (i.e. while having ablution), and one should put it at the highest point in a library and the like.
Among the most important thing in honoring the Mus-haf is to act according to its teachings, and not to abandon it either by not reciting it, or by not referring to it for judgment, or not acting according to it, and not referring to it for Fatwa.
Disdaining the Mus-haf is putting it under the pillow, for example, or putting it in places that are not appropriate for it.
We advise you to refer to the book entitled at-Tibyaan, authored by Imam an-Nawawi for great benefit.
For more benefit, please refer to Fataawa 363280 and 90334.
Allah knows best.