It is permissible for a menstruating woman to recite Qur’aan according to the more correct scholarly opinion, especially if she needs to do that for fear of forgetting (what she had memorised), or to study for a test, or to seek healing thereby, on condition that she does not touch it, because no one should touch it except one who is in a state of purity. If she needs to read from the Mushaf, she may touch it with a barrier, such as a piece of clean cloth or a handkerchief or gloves and the like. 

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: 

It is obvious that women used to menstruate at the time of the Messenger of Allah (blessings and peace of Allah be upon him) and he did not forbid them to recite Qur’aan. He also did not forbid them to recite dhikr (remembrance of Allah) or du‘aa’ (supplication). In fact he instructed the menstruating women to go out on the day of Eid and recite takbeer with the Muslims. End quote. 

Majmoo‘ al-Fataawa, 21/460 

He also said: 

There is no Sunnah at all forbidding (the menstruating woman) to recite Qur’aan. Women used to menstruate at the time of the Messenger of Allah (blessings and peace of Allah be upon him), and if reciting was haraam for them, as prayer is, then this would have been one of the things that the Prophet (blessings and peace of Allah be upon him) forbade to his ummah, and the Mothers of the Believers would have known of it, and it would have been one of the things that they transmitted to the people. As no one transmitted any such prohibition from the Prophet (blessings and peace of Allah be upon him), it is not permissible to make it haraam. It should also be noted that he did not forbid that, and as he did not forbid it, even though there were so many menstruating women at his time, it is known that it is not haraam. End quote. 

Majmoo‘ al-Fataawa, 26/191 

The scholars of the Standing Committee for Issuing Fatwas said: 

It is permissible for the menstruating woman to recite from memory without touching the Mushaf directly, if she needs to recite Qur’aan so that she will not forget it. End quote. 

Fataawa al-Lajnah ad-Daa’imah, 4/232 

Shaykh Ibn Baaz (may Allah have mercy on him) said: 

It is permissible for the woman who is menstruating or bleeding following childbirth (nifaas) to recite Qur’aan from memory, because these conditions last for a long time, and drawing an analogy between them and the case of one who is junub is incorrect. Based on that, there is nothing wrong with a female student reciting Qur’aan, or a teacher doing so for an exam or otherwise, if it is done from memory and not from the Mushaf. But if one of them needs to read from the Mushaf, there is nothing wrong with that on condition that it be done with a barrier. End quote. 

Majmoo‘ Fataawa Ibn Baaz, 6/360 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

It is permissible for the menstruating woman to read Qur’aan from a tafseer (commentary) or otherwise if she is afraid of forgetting what she has memorised. If it is from a tafseer, then it is not stipulated that she should be in a state of purity, but if it is not from a tafseer, rather it is from the Mushaf, then it is essential for her to make a barrier between her and it, such as a handkerchief or gloves and the like, because it is not permissible for the menstruating woman or anyone who is not in a state of purity to touch the Mushaf. End quote. 

Fataawa Noor ‘ala ad-Darb by Ibn ‘Uthaymeen, 123/27 

Shaykh Ibn ‘Uthaymeen was asked about the ruling on reciting Qur’aan from memory for a menstruating woman who is seeking reward or for the purpose of ruqyah as prescribed in Islam. 

He replied: 

If the menstruating woman recites Qur’aan for a purpose other than just reciting, there is nothing wrong with that. If she recites Qur’aan in order to seek healing thereby or as part of a wird for which she is reciting it, or for the purpose of teaching or learning, there is nothing wrong with that, because she is reciting it for a reason. End quote. 

Fataawa Noor ‘ala ad-Darb by Ibn ‘Uthaymeen, 123/21 

Based on that, there is nothing wrong with the menstruating woman performing ruqyah for herself by reciting Qur’aan, du‘aa’s and dhikr that are prescribed in Islam. There is also nothing wrong with her reciting from the Mushaf, on condition that she does not touch the Mushaf except with a barrier. 

And Allah knows best.