As for the question, We have to keep in mind the fact that during their menses, women can read the Qur’an and make dhikr (remembrance of Allah) and du`a’ (supplication), though they are forbidden from fasting, performing Salah (prayer) and touching the Mushaf (copy of the Qur’an). In this way, they still have the opportunity to observe Laylatul-Qadr (the Night of Power) by reciting the Qur’an and making dhikr.

In his response to the question in point, the prominent Muslim scholar, Dr. Muhammad Abu Laylah, professor of the Islamic Studies & Comparative Religions at Al-Azhar Univ. states:

A menstruating woman should not perform Salah (prayer) nor touch the Qur’an, but she can make du`a’ (supplication) and share with other Muslims their prayer by watching and listening to TV channels or radio stations that broadcast Tarawih prayer live.
She can ask someone to put the Mushaf on a table or a stand and read from it without touching it. She can take a cassette and listen to Qur’anic recitation.

Stressing the permissibility of a woman reciting the Qur’an in their menses, we would like to cite the fatwa issued by the Saudi House of Fatwa headed by the late Sheikh `Abdul-`Aziz Ibn Baz (may Allah bless his soul):

There is nothing wrong with a menstruating woman or a woman in post-natal bleeding to recite the Qur’an, because there is no clear-cut authentic Hadith that forbids them from doing so. However, it is reported in an authentic Hadith that one who is Junub (one in a state of impurity following sexual intercourse or wet dream), must not read the Qur’an while he or she is impure, according to the Hadith reported by `Ali (may Allah be pleased with him).

The Hadith reported as regards the menstruating woman and the one experiencing post-natal bleeding reads: “The menstruating woman and the one who is Junub are not to read Qur’an”. This Hadith is reported by Ibn `Umar, but it is Da`if (weak), because the Hadith was reported by Isma`eel Ibn `Ayyaash from the Hijaaziyeen, and he is famous for narrating Da`if Hadiths from them.

However, such woman (in menstruation or post-natal bleeding) should not touch the Mushaf; she can recite from her own memory [or from a copy of the Qur’an without touching it, as per the above opinion]. As for the Junub, he or she is not to recite the Qur’an, whether from memory or from the Mushaf, until he or she has performed Ghusl (purificatory bath). The difference between them is that the time span for the one who is Junub to have himself or herself purified is very short; he or she can do Ghusl right away after lovemaking or wet dream. The Junub does not stay in this condition for long, and it is up to him/her when he/she wants to make Ghusl; if he/she does not find water, he/she can do Tayammum (dry ablution) and then he or she can pray and read the Qur’an. But the woman in menses or in post-natal bleeding has no control over her situation – the matter is up to Allah the Almighty.
therefore, it is permissible for them to recite the Qur’an so that they do not forget it and they do not miss learning the teachings of Shari`ah from the Book of Allah. If that is the case, then it should certainly be permissible for them to read books containing du`a’ that are mixed with verses and Hadiths, etc. This is the view believed to be the most correct.

Finally, Sheikh Muhammad Iqbal Nadvi, Imam of Calgary Mosque, Alberta, Canada, and Former Professor at King Saud University, Riyad, Saudi Arabia, concludes:

A menstruating woman is not supposed to pray. She can do the following:
1. Read as much as she can to increase he knowledge about Islam.
2. Make du’a’ and spend time making dhikr to Allah Almighty.
3. Listen to the Qur’an or read from her memory.
4. Watch Islamic programs or shows on TV or video to educate herself about Islam.
5. Attend religious classes to be always around the committed sisters.