I have traveled to IST as a single female, and have felt perfectly safe - although I will say that a single American female will definitely get some looks. I stuck to the busy/tourist areas, and didn't go out alone at night, and never felt like I was in danger. It's a magnficent city, with lots to do and see. Besides the mosques, my favorite place is the bazaar - great deals, and a fun way to pick up souvenirs. I purchased a tea set and tray that is similar to the kind they use to serve tea to shopping customers, which always give me an opportunity to talk about my travels there when people visit my house

I've also been to Kusadasi and Izmir, which are also beautiful places and have significant historic value, if you have time.