When talking about women and Tora scrolls, one often hears that women may not touch the Tora scroll when they are menstruating. This is a myth and simply not true at all!
First some background: A menstruating women is called "nidda". This is only one category of "Tum’ah" (ritual impurity). Other categories are men who had a seminal discharge ("ba’al keri") or anybody who had contact with a dead body. Tum’ah played a big role in times of the temple where it was relevant to define who can access the temple and bring or eat sacrifices.
So, what does the Talmud say? The Talmud forbids only the ba’al keri from reading Tora, and allows everybody else (Babylonian Talmud Berakhot 22a, Jerusalem Talmud Berakhot 3:4). Some rabbis even reject all attempts to apply the concept of tum’ah to anything outside of the temple. A well-known statement by Rabbi Yehuda ben Beteira supporting this view is "Words of Torah are not susceptible to tum’ah." (Babylonian Talmud Berakhot 22a). Today, even the most orthodox synagogues seem unconcerned that a ba’al keri is called to the Tora (as it is encouraged to have sex on Shabat, on Saturday morning one would assume that most married men are ba’alei keri).
Why then, is there still an issue with nidda women? Somehow the custom developed that women who are nidda don’t enter the synagogue or pray. Still, it is clear that this is only a custom, not halacha. Some commentators look favorably on this custom, some don’t. Two example texts:
Any impure person, even [a woman in] a niddah state or a gentile, may hold a Torah scroll and read it. The words of Torah do not contract ritual impurity. This applies when one’s hands are not soiled or dirty with mud. [In the latter instance,] one should wash one’s hands and then touch the scroll. [Rambam, Mishne Tora, Laws of Tefillin, Mezuzah, and Sefer Torah 10:8, see also Shulhan Arukh, Yoreh De’ah 282:9]
There are those who wrote that a woman in niddah while in the days of her bleeding should not enter the synagogue, pray, mention God’s name, or touch a Torah scroll (Haggahot Maimoniyot). And there are those who say she is permitted to do all of the above and this is correct (Rashi, Hilkhot Niddah). However, the custom in these countries follows the first opinion. [Rema on Shulchan Aruch, Orach Chayyim 88:1]
So the question of touching the sefer Tora is tied with entering the synagogue and praying. The idea that women who are nidda do not pray is highly problematic. Women may not have the same obligations in prayer as men (to pray three times daily with a minyan), but I think we pretty much all agree that women do have the commandment to pray daily. Also they are commanded to say blessings before eating and birkat haMazon like men. So I think nobody today says that women should not pray or make blessings when they are menstruating.
With regard to not entering the synagogue and other things that have been mentioned in that context (standing in front of another women, answering amen to prayers, looking at the Tora, or entering cemetaries) – I don’t see that anybody is much concerned about this nowadays. The only thing that appears to have stuck is the thing about touching Tora – even thought the Talmudic prohibition of a (by definition male) ba’al keri is not applied.
Bottom line: There is no concern about a nidda woman touching the Tora. At all.
Sources and longer discussion: Ta Shma Guide: May Women Touch A Torah Scroll?.