I'm going to pretend I'm not cheating by naming my three favorite AH fics and my three favorite vampire fics.
1. Ithaca is Gorges by giselle-lx (canon, New Moon from Edward and Carlisle's perspectives, found on Ao3, FF, and the author's website). While you're on her page, read her one shot called 24601, which is from Jasper's POV and is the best one shot in the fandom IMO.
2. Goodbye Peter Pan by Bled Dry (New Moon AU in which Edward and Bella are reunited ten years after he left, found on FF)
3. Someone Else's Memoirs by a certain slant of light (AU, Jasper/Bella that is largely about grieving, very sad but beautifully written, found on FF)
1. The Blessing and the Curse by The Black Arrow (has a touch of the supernatural in that Edward can read one mind, and it's Bella's; dark and broody, found on FF)
2. Wide Awake by angstgoddess003 (Edward and Bella are teenagers with traumatic pasts who deal with their insomnia together, a Twific classic, found in the files here)
3. Love Between Sexes by EFC (Edward and Jasper were best friends in college and are reunited ten years later and finally admit they were more than friends. Lots of angst, lovely love story, found on FF)