One fellow (I assume it's a bloke from the avatar) exploring more unusual uses for the Twitter medium is @VeryShortStory
Each of his tweets is a self-contained story, usually in the form of a set-up, followed by a twist. They are uniformly clever and often both sad and funny in equal measures.
If you want to write your own very short story on Twitter, follow your story with the hashtag #vss, that way people can search for it. That's exactly what I did a couple of months ago:
The silver harmony of angel-song soaks His kingdom, gentles the ears of the good and the wise. Please, God, when will it stop? #vss
I'll probably write more in the future; it's a good exercise if nothing else. I'm also thinking of using this particular one in a sound-art project soon--watch this space. In the meantime, I felt urged to share @VeryShortStory's splendid work. If you're on Twitter, follow him.
P.S.: I also enjoy the tweets of @trapphic, who writes humorous sci-fi VSSes. Go follow him too. If you know of anyone else doing similar stuff, share them in the comments.