New Orleans has hosted the Super Bowl, Final Fours, championship boxing bouts, presidents, a pope, the Dalai Lama and coming in September: The 2013 National Beard and Moustache Championships(Sept. 6-7), the Westminster show of male facial hair. There are 18 categories to medal in, and anything goes except artificial hair.
Germany is home to the World Beard and Moustache Association, which held its first competition in 1990. National Geographic has some good pictures of 2010 competitors. Britain's club has its own impressive competition.
In the U.S., regional groups are cropping up and there are several regional competitions. There is a New York competition, which allows women to participate (and Coney Island has its own competition). There's also a New York beard alliance. Los Angeles has a competition. Missouri has an active hirsute community. There's a club in Philadelphia. Austin. Houston. Bonnaroo has a beard competition this year, and it's got a category for fake beards. The online Beard Club has a links page with a round up of tumblr sites with photos, how to sites, competitions and more.
Update: There also is a local group for those who want to stop shaving and start networking.
The championships in New Orleans are open to men only. There are three main divisions with subcategories: Mustache (Salvador Dali, Hungarian, Imperial, freestyle, etc.), Full Beards (natural, Geribaldi, freestyle) and Partial Beards (Fu Manchu, Muskateer, Amish, sideburns, freestyle, etc.). Competitors are allowed to use various hair products, but from competition photos, it seems that some sort of period costume is a good idea.