I've written before that I'm ambivalent about screenplay contests, but there's no doubt that winning one of the major ones can give your career a boost. If you're interested in seeing what contests are out there for you, check out the website www.screenplaycontests.com.
At this writing, they have information about the Nickelodeon Writing Fellowship, the DC Shorts Screenplay Competition, the Fresh Voices contest, and many more. Just to give you a quick recap of my guidelines about contests:
1. Make sure the prizes are substantially more than the entry fees;
2. Check the history of the contest and get in touch with previous winners to find out what they think now about their experiences (you can probably find them via Facebook or using Google);
3. Make sure that you're not giving up your copyright by entering or by winning;
4. Make sure that if you win your script is not tied to the sponsors in a way that doesn't serve you (for instance, winning $500 is not a good trade for giving someone an exclusive option on your script for a year or two unless you feel very confident in their ability to get the script made);
5. Make sure you have read all the fine print. If any of the clauses confuse you, get clarification.
Good luck!
(There's a lot more good information about screenwriting at my other blog: www.ScreenwritingSuccess.com)