The Warren Adler Contest-- April 11 end date, $15 fee. This contest is devoted to promoting the short story as a valid art form.
Writer's Weekly 24-Hour Contest-- 24 April. Yes, that's right. You will have 24 hours to write and submit a short story on the topic to be posted on that date.
Jerry the Jazz Musician Short Story Contest 31 May. I don't actually know what it has to do with Jazz, or who Jerry is.