Crime in Reykjavik is basically non-existent. Even petty thieves are only rarely seen. Violent crimes? Not often here. There are no safety issues concerning drugs or homeless people. The only area in Reykjavik that a single woman may not want to visit late at night is Austurvöllur Park, and that's only because it's a popular place for drunk people, who tend to like to keep to themselves anyway. If you want safety, Reykjavik is the perfect destination for you.