A friend recommended a site called Short Story America (as a market for short stories). Seems like a nice enough place. (For some reason my browser window gets resized when I go to that url, but otherwise it's fine.)
I couldn't find payment information, but the editor, Tim Johnston, replied promptly to my query and told me they pay a flat $100 for short stories, $50 for flash fiction. (Though it occurs to me now I don't know at just what wordage they make the distinction.)
I submitted a 1,000 word story. Will report back on any results.