Right now, I'm working on a story about Newt Scamander and a group of his friends. Although I've pulled all of the main characters from canon, their personalities are essentially original. Along with Newt, I'm writing about his wife, Porpentina; Professor Kettleburn, who gets one mention in Prisoner of Azkaban; and Jocunda Sykes, who appears in Quidditch Through the Ages as the first person to cross the Atlantic on a broomstick. Although I've been reluctant to claim them as original, since I can't take credit for at least part of their names, there's not very much I haven't made up about them.
To address the topic of self-insertion, I'm finding it impossible not to commit that act. I don't have one particular self-insert in my little Scamander universe, though. There's a lot of myself in my characters, particularly Newt and Porpentina, and although none of them acts for me--nor do I want them to do so--I know I wouldn't be writing this story if it weren't for experiences I've had. Namely, Newt's unhappiness with his first Ministry job wouldn't have appealed to me if I hadn't had a negative college experience and dropped out. Similarly, Porpentina wouldn't be a writer with nothing to write about if I hadn't felt that way. (If only she had fanfiction, too!)
This is a much longer introduction than I'd intended, so I'll stop now. I'm looking forward to reading everyone else's posts here.