Are menstrual cups safe?

Yes. Yes, they are safe to use, and even "may save a few trees".
Should be. There is no evidence to the contrary. The most important thing about menstrual cups is that you use them as directed and you follow closely the recommendations for cleaning and storage. They can generally be worn up to 12 hours before emptying, cleaning, and re-inserting. So far, there are no reported cases of toxic shock syndrome. They are an environmentally responsible alternative to tampons.