Short answer
It is disliked to put the mushaf directly on the floor without need, because it goes against the honour due to it. Put it on a stand, a table, a cushion or your lap. Carrying it in a bag below your waist is fine; keep it out of your back pocket and away from your feet.
Placing the mushaf on the bare floor without need is makruh, as it is against the respect (ta'zim) it is owed; a stand, a raised place or a clean cloth is the proper way. Stretching the feet toward it, putting other things on top of it, or sitting on it are disrespectful and to be avoided.
The four schools agree on this.
Explanation
Allah says that honouring His symbols comes from the piety of hearts (22:32). When the Jews of Madinah brought the Torah to the Prophet ﷺ, he took the cushion from under himself, placed the Torah on it, and said he believed in it and in the One who sent it down. If he honoured the Torah this way, the Quran deserves at least as much. The four schools agree on this respect.
- In the mosque: hold the mushaf in your hands or lap, or use a wooden stand (rehal). A stand on the floor is fine.
- Below the waist: carrying it in a backpack, a handbag at your side or the boot of a car is fine, since it is in a container and you are carrying it, not lowering it.
- Pockets: a small Quran in a shirt or front pocket is fine; avoid a back pocket where you would sit on it.
- Stacking: put the Quran on top of other books, not under them.
- Feet: do not stretch your legs toward it while it is close and at your level.
If the mushaf slips and falls, pick it up and carry on. Nothing is owed, though some people give a little charity out of love for it. The Hanafi texts also allow kissing the mushaf as a sign of respect. For worn copies you can no longer use, see how to dispose of an old Quran.
The evidence
Quran, Surah Al-Hajj 22:32
ذَٰلِكَ وَمَن يُعَظِّمْ شَعَٰٓئِرَ ٱللَّهِ فَإِنَّهَا مِن تَقْوَى ٱلْقُلُوبِ ﴿٣٢﴾
That [is so]. And whoever honors the symbols of Allah - indeed, it is from the piety of hearts.
Honouring what Allah has made sacred, the mushaf first among them, is a sign of piety.
Sunan Abi Dawud 4449
Graded hasan... آمنت بك وبمن أنزلك ...
"He removed the cushion from under himself, placed the Torah on it and said: I believe in you and in the One who sent you down."
He raised a revealed book onto a cushion instead of leaving it low, which shows the respect owed to scripture.
Where the ruling is recorded
- al-Fatawa al-Hindiyya, Kitab al-Karahiyya, chapter on reciting the Quran and the manners of the mushaf. Lists acts of disrespect toward the mushaf, such as placing it low, putting things on it and stretching the feet toward it.
- Radd al-Muhtar (Ibn Abidin), Kitab al-Hazr wa al-Ibahah, discussion of honouring and kissing the mushaf. The respect owed to the mushaf and the permissibility of kissing it.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.