Assess your child's voice in terms of the kinds of songs he is best suited for. If the singing contest is along the lines of an open mike in which he can perform whatever he wants, no problem. If the objective is to find promising young voices for light opera, however, and he's only comfortable doing hip-hop, it's going to be both a waste of the judges' time and a disappointment for your child when he gets rejected. You'll also want to take into consideration whether your child's voice is a better match for recording studios or for actual acting. Many a singer--regardless of age--may hit all the right notes but be unable to get into character, express emotion and move all at the same time.
Subscribe to free services such as SingingAuditions.net, StageDoorAccess.com and AuditionsSinging.com to receive the latest notices of upcoming tryouts in your area. Check the newspapers regularly for audition and contest notices. If there's a local music academy, query whether it holds singing competitions and, if so, ask to be put on its mailing list.
Review the contest rules carefully, especially regarding eligibility to participate. Some contests, for instance, may require that you bring proof of your child's age (i.e., a birth certificate). They will usually request a professional headshot of the child and a brief summary of training and experience as a singer/performer. (The younger the child, the less likely they'll expect the latter.) Respect the deadlines for registration; contest slots for children tend to fill up very fast. If you wait until the last minute to sign up, your child might not be allowed to sing.
Find out whether your child will be expected to sing a song from a specific list (or, in the case of a theatrical production, a song from that show) or can choose her own song. If the contest material has to be from a designated list, you'll need to acquire a copy of the sheet music for practice. If it's a song of her own choice, you'll need to have a copy of the sheet music to give to the accompanist on the day of the contest/audition.
Tape record the music for practice. (If you don't play the piano yourself, you'll need to find someone to do this for you.) Many parents make the mistake of believing that a child's ability to sing along with a Broadway soundtrack is a guarantee of success. What they don't take into account is that (1) the key the song was recorded in may not be the same as what's on the commercial sheet music, and (2) the more a child gets accustomed to the sound of a full orchestra, the more it's going to throw off his timing when his only accompaniment at the contest is a piano.
Videotape your child's at-home rehearsals so she can learn how to improve her body language, eye contact and hand movements. (Personality and self-confidence are evaluated by judges just as much as the actual singing.)
Arrive early for the contest and follow whatever instructions you're given.
Respect the decision of the judges. (You'd be surprised how many parents go ballistic when their children are passed over for a prize.)