Fair Isle is actually now totally owned by the National Trust. I could be wrong, but I don't think the Shetland Council has anything to do with administering it. However, having said that, I would think that Shetland libraries and museums would be a good first place to search out historical and genealogical information from Fair Isle. I believe there was also a bit of movement from Fair Isle to the north isles of Orkney (Sanday, Stronsay).