Praise be to Allah.
It is haraam for a menstruating woman to stay in the mosque, because of the report narrated by al-Bukhaari (974) and Muslim (890) from Umm ‘Atiyyah (may Allah be pleased with him) who said: He -- meaning the Prophet (blessings and peace of Allah be upon him) -- commanded us to bring out on the two Eids the girls who had attained puberty and those who were in seclusion, but he told the menstruating women to keep away from the prayer-place of the Muslims.
So the Prophet (blessings and peace of Allah be upon him) forbade menstruating women to come to the Eid prayer place, and he told them to keep away from it, because it comes under the same rulings as the mosque. This indicates that a menstruating woman is not allowed to enter the mosque. This is the view of the majority of scholars.
There is no distinction between the mosque in which Jumu’ah prayer is held and one in which it is not held, or between one which has a minaret and one which does not. What matters is the fact that it is a mosque, i.e., a place that is devoted for prayer.
With regard to prayer rooms in workplaces and schools, which remain the property of the owner and are not established as mosques per se, they do not come under the same rulings as mosques, so it is permissible for a menstruating woman to enter them and stay in them.
The scholars of the Standing Committee for Issuing Fatwas were asked:
Does the prayer room in the university residence come under the same rulings as a mosque with regard to praying tahiyyat al-masjid (two sunnah rak’ahs performed upon entering to “greet the mosque”) and reciting the adhkaar for entering and leaving the mosque?
Prayer rooms in university residences and elsewhere do not come under the same rulings as mosques in every aspect. End quote.
Fataawa al-Lajnah al-Daa’imah (5/170).
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked about prayer rooms in schools that are set up for Zuhr prayer only: is it permissible for a menstruating woman to enter them?
Prayer rooms in schools do not come under the same rulings as mosques. Rather they are prayer rooms. Not every place in which prayers are held is regarded as a mosque. The mosque is that which is set up for prayer in general and is built and prepared for that purpose. Simply taking a place as a place in which to pray does not make it a mosque.
Based on that, it is permissible for a menstruating woman to enter and stay in the prayer room of a school. End quote from Liqa’ al-Baab al-Maftooh, 22/27.
To sum up: it is not permissible for you to sit in the mosque when you are menstruating, but it is permissible to do so in a prayer room that does not come under the same rulings as a mosque.
And Allah knows best.