The University of Nevada Las Vegas maintains a writing competition web site which I've linked this morning, here in this blog post, and at the side bar. Because the 2008-09 academic year is just getting underway, many currently advertised competitions appear out of date, but competitions are a yearly event, and this year's deadline will approximate last year's. Taking a look at the competitions now makes sense, as you plan out your year and your writing topics for law journal activities, seminars, or independent study papers.
While many of the competitions describe "essays," in most cases its a paper with footnotes that is the focal point. The page lengths for the papers run between 25 and 50 pages of double spaced text. Every competition is a little bit different. Take a look at the general advertising information for all the competitions, and for the ones that interest you, also take a look at the competition rules, usually set out in a discrete location. As with deadlines, the previous year's competition rules will provide a good idea of what this year's competition rules will be. Also, in the idea-generation stage of choosing your writing topic, the winning titles will give you a sense of the possibilities.