Harper's Bazaar are running a free to enter short story competition and offering a two night stay at a hotel in Hampton Court. When I say it's free I do of course mean there's no entry fee, but this one is free in another way – you'll be giving your story away for free simply by entering!
I strongly advise all writers NOT to give up all rights to their work, even if they get paid for them. To give up all rights for no reward at all seems like an even worse idea.
This free novel writing competition by the similarly named Harper Collins (no relation!) is considerably better in that the author retains their copyright. The prize is an editorial review and editorial mentoring. They're not guaranteeing to publish the winning stories, but it does seem as though there's a reasonable chance of that happening.