Can You Read the Quran during Your Period?
Question

Is it permissible to read the Quran while a woman is having menses (periods)?
Praise be to Allah.Reading the Quran during menses
Reading the Quran during menses is one of the issues on which the scholars, may Allah have mercy on them, differed.
The majority of fuqaha say that it is haram for a woman to recite Quran during her period, until she is tahir (pure) again. The only exceptions they make is in the case of dhikr (remembrance of Allah) and phrases that are not intended as tilawah (recitation), such as saying “Bismillahi’r-Rahmani’r-Rahim” or “Inna Lillahi wa inna ilayhi raji’un”, or other phrases from the Quran which are repeated as general du’as.
Evidence for forbidding women from reading the Quran during menses
They base their evidence for forbidding menstruating women to recite Quran on several things, including the following:
1- Menstruation is seen as coming under the rulings that apply to one who is junub (in a state of impurity following sexual intercourse), because both states require ghusl. This is based on the hadith narrated by ‘Ali ibn Abi Talib (may Allah be pleased with him), according to which the Messenger of Allah (peace and blessings of Allah be upon him) used to teach the Quran and he never prevented anyone from learning it except those who were in a state of janabah (major ritual impurity).” (Reported by Abu Dawud, 1/281; al-Tirmidhi, 146; al-Nasai, 1/144; Ibn Majah, 1/207; Ahmad, 1/84; Ibn Khuzaymah, 1/104. Al-Tirmidhi said: a sahih hasan hadith. Al-Hafiz ibn Hajar said: the truth is that it is the type of hasan hadith that could be used as evidence).
2- The hadith of Ibn ‘Umar (may Allah be pleased with them both), according to which the Prophet (peace and blessings of Allah be upon him) said: “The menstruating woman and the one who is in a state of major ritual impurity (janabah) should not recite anything of the Quran.” (Reported by al-Tirmidhi, 131; Ibn Majah, 595; al-Daraqutni (1/117); al-Bayhaqi, 1/89. This is a da’if hadith, because it was reported by Isma’il ibn ‘Ayyash from the Hijazis, and his reports from them are da’if as is well known to those who are conversant with the study of hadith. Shaykh al-Islam Ibn Taymiyah said (21/460): it is a da’if hadith by the unanimous agreement of the scholars of hadith. See Nasb al-Rayah, 1/195; al-Talkhis al-Habir, 1/183).
Evidence for allowing women to read the Quran during menses
Some scholars say that it is permitted for a menstruating woman to recite Quran. This is the opinion of Malik, and one opinion narrated from Ahmad, which Ibn Taymiyah preferred and which al-Shawkani believed to be correct. The scholars based the following points on this opinion:
1- The principle is that things are allowed and permitted unless there is evidence to the contrary. There is no such evidence to say that a menstruating woman is not allowed to recite Quran. Shaykh al-Islam Ibn Taymiyah said: “There is no clear, sahih text to indicate that a menstruating woman is forbidden to recite Quran… It is known that women used to menstruate at the time of the Messenger of Allah (peace and blessings of Allah be upon him), and he did not forbid them to recite Quran, or to remember Allah (dhikr) and offer du’a.”
2- Allah, may He be glorified and exalted, commands (Muslims) to recite Quran. He praises the one who does so, and promises him (or her) a great reward. No one is excluded from this except the one concerning whom there is solid evidence (dalil), and there is no such evidence in the case of menstruating women, as stated above.
3- The analogy between the menstruating woman and the one who is in a state of janabah is made despite the fact that there are differences between them. The one who is in a state of janabah has the option of removing the “barrier” by making ghusl, unlike the menstruating woman. A woman’s period usually lasts for some length of time, whereas the person who is in a state of janabah is required to do ghusl when the time for prayer comes.
4- Preventing a menstruating woman from reciting Quran deprives her of the chance to earn reward, and it may make her forget something of the Quran, or she may need to recite it for the purposes of teaching or learning.
From the above, it is clear that the evidence of those who allow a menstruating woman to recite Quran is stronger. If a woman wants to err on the side of caution, she can limit her recitation to the passages which she is afraid of forgetting.
Touching the Quran during menses 
It is very important to note that what we have been discussing here is restricted to what a menstruating woman recites from memory. When it comes to reading from the Mus-haf (the Arabic text itself), a different rule applies. 
The correct view of the scholars is that it is forbidden to touch the mus-haf when one is in any kind of state of impurity, because Allah says (interpretation of the meaning): “… which none can touch except the purified.” [al-Waqi’ah 56:79]
In a letter to ‘Amr ibn Hazm, the Prophet (peace and blessings of Allah be upon him) told the people of Yemen: “No one should touch the Quran except one who is tahir (pure).” 
The hadith is reported by Malik, 1/199; al-Nasai, 8/57; Ibn Hibban, 793; al-Bayhaqi, 1/87. Al-Hafiz ibn Hajar said: A group of scholars classed this hadith as sahih because it is so well known. Al-Shafi'i said: It is proven by them that it was a letter sent by the Messenger of Allah (peace and blessings of Allah be upon him). Ibn ‘Abd al-Barr said: “This letter is famous among the scholars of sirah, and is so well known among the scholars that it does not need an isnad. It is like tawatur because the people accepted and recognized it. Shaykh al-Albani said that it is sahih. Al-Talkhis al-Habir, 4/17. See also: Nasb al-Rayah, 1/196; Irwa al-Ghalil, 1/158).
(Hashiyat Ibn ‘Abidin, 1/159; al-Majmu’, 1/356; Kashshaf al-Qina’, 1/147; al-Mughni, 3/461; Nayl al-Awtar, 1/226; Majmi’ al-Fatawa, 21/460; al-Sharh al-Mumti’ li’l-Shaykh Ibn ‘Uthaymin, 1/291)
For more about issues related to menses, please see these answers: 72417 , 75727 , and 121366 .
And Allah knows best.