Interestingly, a really large number have unhappy endings. I think there's really something to the resonance of the lost love idea... I mean, Shakespeare wrote lots of happy endings, but no one mentions those couples. It's always Romeo and Juliet instead. (Though a lot of the happy couples [i.e., Beatrice and Benedick] seemed more believable to me than Romeo and Juliet, but I digress...)
Still, I'm happy to see some favorites of mine on here... Casablanca, The Princess Bride, Beauty and the Beast, Titanic (yes, I liked Titanic, and I'm not afraid to admit it ;) ).