Case: During Menses, it is not unlawful to celebrate Allah’s Praises (Saying Takbeer, Tasbeeh, and Tahmeed), saying Bismillah before acting upon something (eating, drinking, etc..), reading Hadeeth, Islamic discussions, invocation upon Allah (Dua’), and listening to the recitation of Qur’an.
It has been confirmed by ‘Aisha (radhi Allaahu anha) that the Prophet (صلى الله عليه و سلم) used to: “lean on my lap and recite Qur’an while I was in menses” (Al-Bukhari and Muslim).
Also Umm ‘Atiyya reported that she heard the Prophet (صلى الله عليه و سلم) saying: “The Unmarried young virgins and the mature girl who stay often screened or the young unmarried virgins who often stayscreened and the menstruating women should come out and participate in the good deeds as well as the religious gathering of the faithful believers but the menstruating women should keep away from the Musalla (praying place).” 
Case: The reciting of the Qur’an by a menstruating woman either via direct looking or silently (in her heart) without the actual uttering by the tongue is permissible.
The majority of the scholars are with the opinion that it is not prohibited for her to utter the Qur’anic recitation. Imam Bukhari, Ibn Jareer At- Tabaree and Ibn Al-Munthir, Imam Malik, and Imam Ash-Shaafi’ee (in one of his earlier sayings) are with the opinion that it is permissible for her to recite the Ayah (verse in the Qur’an).
Shayekul Islam Ibn Taymeeyah said: “Basically, there is no Sunnah to prevent her from reciting Qur’an. The “hadeeth” that: “A menstruating woman and the one who is in state of Jannabah  should not recite anything of the Qur’an”, is a weak hadeeth as agreed upon by the knowledgeable scholars of Hadeeth.
In fact women at the time of the Prophet (صلى الله عليه و سلم) used to have their monthly cycles of menses.Had the recitation of Qur’an been prohibited for them (as it is the case with Salat) then the Prophet (صلى الله عليه و سلم) would have made it clear to his Ummah (Islamic community) and his wives would have known about it and they would have dissipated this knowledge to the people. Because no one had related a prohibition by the Prophet (صلى الله عليه و سلم) about this issue, then it cannot be made unlawful especially when it is known that he (صلى الله عليه و سلم) did not prohibit it at the time when menses was very much present.
Accordingly, recitation by the tongue is not Haram (legally unlawful). This author said: “having known the dispute between the scholars about this matter, then it is more proper for the menstruating woman not to recite Qur’an by the tongue except when it is necessary e.g. if she is a teacher or if she is preparing for an exam.”
 Takbeer: saying Allahu Akbar (Allah is The Greatest); Tasbeeh: saying Subhana Allah (Most Glorified is Allah); Tahmeed: saying Al Hamdulil’lah: All Praise is due to Allah.
 Saheeh Al-Bukhari (Arabic-English) V.1, P.179, Hadeeth # 296.
 Agreed upon Hadeeth. In Saheeh Al-Bukhari (Arabic- English), V. 1, P. 192, Hadeeth # 321.
 Jannabah: Is the state in which a man (or a woman) has sexual discharge during sexual intercourse or in a dream, etc. In this state a Muslim is not allowed to pray or do other kinds of worships unless and until he (she) performs Ghusl (have a full shower) or do Tayammum if he/she cannot use water for any reason.
 Ibin Taymeeyah in Al-Fatawa (religious decrees) V. 26, P. 191.
Source : [eBook] Natural Blood of Women – Shaykh Uthaymeen – Translated by Dr Saleh As-Saleh