Iceland is easily one of the world's safest countries and crime is almost non-existent here. However, depending on the area in Iceland you're traveling to, make sure to keep the following in mind:
Crime in Reykjavik is basically non-existent, even petty thieves are only rarely seen. Violent crimes? Not here. There are no safety issues concerning drugs or homeless people. The only area in Reykjavik that a single female may not want to visit late at night is Austurvöllur Park - and that's only because it's a popular place for winos, who like to keep to themselves anyway. If you want safety, Reykjavik is the perfect destination for you.
The rural areas of Iceland are safer than you could possibly imagine. The only risks to your safety here are slower response times by emergency assistance services in remote locations of Iceland. If you are driving in Iceland, carry a flashlight with you, keep your headlights on, and don't go off-road.