I find that twilight stories are a guilty pleasure of mine.
My favorite characters are (in no particular order): Bella, Angela, Rose, Alice, Jasper, Renesmee, Seth, Charlie, Carlisle, Jane, and Alec. Most stories involving any of those characters will catch my eye.
I find myself most drawn to stories in which the Cullen's are vampires, or at the very least the supernatural is included. For me their nature is too much a part of their characters to write a plot without it.
Rare and loved stories indeed are those that tell a human story and plot, but keep the Cullen's vampires. Two such stories serve as an examples: Chandler story EverClear and caligula42 story Irritable Grizzly Adams; two of my favorites. The fact that they're vampires is utilized, but the plot is not obsessed with it.
I also enjoy AU stories that move in exciting and epic scope. Unloved Unwanted by content1 is a great example. Her expansion of the macro vampire world and elaboration on the Volturi lifestyle and characters are great.
I'm a huge fan of stories that imaginatively expand on Bella's shield rather then make her have her stealing all the powers around her. I think my favorite such enhancement was GiveUsAKiss413 story Past and Present up to but excluding the stunning finale. My perspective was forever altered by her imaginative expansion of the concept of a mental shield and I find it difficult to see it not following that logical path were she to continue to work with it.
You could infer from my story selections that I'm "Team Edward". You would be right to a point; it need to be pointed out though that I never really liked him until the middle of Breaking Dawn. What ruins it for me is my belief that a healthy romantic relationship is about mutual support not one-sided control. He makes a great bodyguard but a lousy boyfriend.
The reason I'm not "Team Jacob" is simply because I never saw Bella and Jacob as a realistic healthy pairing. Stephani made it clear from Bella's perspectives that she was not into him that way. It might have changed later given the chance, but it didn't in the stories. The way Stephenie described Jacob's stealing of a kiss from her, the scene where she sprained her hand punching him, makes that quite clear. In the movie, Edward's reaction mirrored my own.
I'm aware that many people do not like the events involving Renesmee. I find that unfortunate as she is one of my favorite characters in the canon. The fast growing, stunningly mature little hybrid with the cool communicative power was great. That said, while I like Jacob's character, I am not a fan of the imprinting. I'd be a fan if she grew up and they fell in love, that would be awesome. Imprinting from the cradle... not so much.
My fanfiction Handle and Page is: Rahveryn