Please let me know when you find interesting contest pages, or journals accepting submissions, so I can add them to this list.
Glimmer train’s current contest runs until the new year: http://www.glimmertrain.com/fictionopen.html
And for poets: http://classicalpoets.org/2012-poetry-competition/ appears to do the same (it’s free!)
For women writers: http://www.wow-womenonwriting.com/contest.php
On the Premises: http://www.onthepremises.com/current_contest.html
More contests which may have ended, but the sites will probably accept submissions again as time goes on: