Leith is a pretty good bet then, just choose carefully as some bits are still well dodgy. It's pretty obvious when you're looking at flats which areas are ok. A little further west and nicer but more expensive is Trinity, and further west again but more central is Comely Bank, which I really like. Right next to the wonderful Stockbridge but much cheaper. South side you have Newington, which is quite studenty, or Bruntsfield which can be pricey but is a fantastic area. Avoid places like Dalry, Pilton, Granton, Gracemount. Have you looked on Zoopla? I find that's a great site for finding flats!