Apex is having a flash fiction Christmas contest. The deadline is December 16.
Unlikely Story has two open submissions. One for people with a SFWA-qualifying publication, to write "the worst story you can bear to see published with your name under the title." Flash only, the deadline is December 15. The other is for a story containing some kind of map, and the deadline for that one is February 1, 2014.
The Jim Baen Memorial Contest is now open. They want near-future hard SF where Tech Is Good. The deadline is February 1, 2014.
Of course, the quarterly deadline for Writers of the Future is nearly upon us. Get your entry in by December 31.
There are various anthos calling for various things.
I'm mulling a second story for the Apex Christmas thing, I've sent my Terribad story off to Unlikely Acceptances, and I'm outlining a pair of map stories and a near-future SF story. And I'm deciding what to send to WotF.