Very, very rarely. I think the only fic I've written that was actively inspired by someone else's fanwork is my TS story Coming To Terms, and that wasn't the story itself, but a specific piece of non-canonical Jim/Carolyn backstory which I used as my emotional center.
I stole that, with permission (since she's one of my betas and a very good friends), from feliciakw's fic Said and Done, which I cannot find for the life of me. Fee, I know your old site is down, but I thought you had that up on your LJ or somewhere, and I can't find it. It should be online, because it's very good. (ETA: now with current link!)
21 – Sequels – Have you ever written a sequel to a fic you wrote, and if so, why, and if not, how do you feel about sequels?
22 – Have you ever participated in a fest or a Big Bang? If so, write about your favorite experience in relation to one. If not, are there any you've thought about doing? And if not, why not?
23 – When you post, where do you post to? Just your journal? Just an archive? Your own personal site?
24 – Betaing – How many betas do you like to use to make sure there aren't any major flaws in your fic? Do you have a Beta horror story or dream story?
25 – Music – Do you listen to music while you write? Do you make playlists to get into a certain "mood" to write your fic? Do you need noise in general? Or do you need it completely quiet?
26 – What is the oddest (or funnest) thing you've had to research for a fic?
27 – Where is your favorite place to write, and do you write by hand or on the computer?
28 – Have you ever collaborated with anyone else, whether writing together, or having an artist work on a piece about your fic?
29 – What is your current project or projects?
30 – Do you have a favorite fic you've written? What makes it your favorite? And don't forget to give us a link!