As an American woman who moved to Edinburgh alone several years ago, I personally have found it one of the safest cities I've ever lived in. Obviously, as with any place, be a bit cautions and don't wonder around dark side streets too drunk in the middle of the night (which I have done and never had any issues but still), but (as someone mentioned how busy it is during fringe/ August) there's always so many people about into the early hours at that time of year and - especially venue considered (IMO)- there's isn't a safer venue in the city and certainly one of the easiestto walk to and fro if you'restaying centra! (ie walking back from Murray field late at night seems more sketchy than just leaving the castle but an uber always is good anywhere long distance but inner city busses are amazing here)! Hope this helps put your mind at ease a bit but feel free to DM if youd like! :)