Many festivals have them. The Woody Guthrie Free Folk Festival, The Kerrville Music Festival, Wildflower Music Festival, Rocky Mountain Folks Festival, Telluride Bluegrass Festival are among the many and might be the most prestigious. Many other smaller ones.
Another big one in the John Lennon Song Contest. Easy to find on-line. The smaller the festival the better the chance of getting a full second look.
I've been a finalist in about 12 or so. They can be fun, most of the festivals are for "Performing Songwriters". Others are geared towards just "the song." You pay's your money and you take your chances."
Look for local songwriting groups. In California we have the West Coast Song Writers. They go from Los Angeles up to Portland that I know of. They have monthly events along with a yearly finals and also just a song contest.
http://www.westcoastsongwriters.org/...l-song-contest
This year a couple of Breedlove's are the for the winners. One of the sponsors here provided the guitars.
To quote Guy Clark.
"Well, I’ve never thought of songwriting as a competitive sport. Everybody supports and inspires one another, but it’s not a competition."