Is there anything prohibiting/restricting a woman from reading Pirkei Avot or the Shulchan Aruch?
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Women can lean definitely Pirkei Avot & Shulchan Aruch (why not?). On the contrary - women are obligated to know their halachic obligations and there is no other way for that other than studying Torah.
Historically, there were many wome who studied Torah. Currently, we have women who teach Torah, and Rabbi Avi Weiss has given some women the title "Rabbanit". Women are allowed to be mashgichot (kosher supervisors), so, I assume that they are learning some parts of Yoreh De'ah to know what to do.
I am citing these as examples of what appears to be a permissibility. However, there are so many differing opinions on this subject, that are all credible. You may want to read this article which throughly addresses the diferent opinions on this issue. I decided not to excerpt anything from it, as I think all the opinions mentioned are valid, and I can't decide which ones I like best.