Is it permissible to browse the Qur’an in the mosque and recite from memory what one has memorized, for someone who does not know how to read or write, in order to avoid falling asleep while awaiting the prayer?

No Marfu' Hadith has been authentically reported regarding the virtue of looking into the Mus'haf (Qur'an), but it has been authentically reported from Ibn Mas'ud (may Allah be pleased with him) that he said: "Continue to look into the Mus'haf," and what is meant by "looking" is reading. Many scholars have explicitly stated that reading from the Mus'haf is better than reciting from memory, because it employs both the tongue and the eyes, thus being two acts of worship. As for reading the Mus'haf without looking, or turning the pages without looking, we see no objection to it as long as it does not reach the level of aimless tampering. What indicates the virtue of reciting from memory has been reported, such as the saying of Abu Umamah: "Recite the Qur'an, and do not be deceived by these hanging Mus'hafs, for indeed Allah will not punish a heart that has comprehended the Qur'an." The preference varies according to circumstances and individuals. For a proficient memorizer, reciting from memory would be better, while for someone who does not memorize well, reading from the Mus'haf is preferable.