Most of these contests are open to children from primary and middle school -- the primary age range of Stone Soup's readers. As most of these contests also post videos of contest winners you can also use this listing to find wonderful music written by children. If you know of other contests, then please post the link in the comments section.
Open to young musicians ages 13 to 18 who are U.S. residents. The prize is $500 plus your piece is performed by an orchestra. The website includes dozens of recordings of winners back to 2007 under "Listen" link in the menu bar. We find the download option the most practical on Macs. The site also includes tutorials to help young composers with orchestration. We highly recommend "Overture di Festival" by Nathan Blair, age 13, as a good piece of music to start listening to.
A British organization devoted to European music prior to the Baroque period (roughly 1600). There is an under-eighteen category, but most of the composers are are in their late-teens to early twenties. This site includes recordings and resources. If you are teaching Shakespeare, then site offers a way to get into the music of the period.
Since 1998 the BBC Inspire Young Composers Scheme has upheld the BBC Proms’ commitment to new music by bringing together young composers, aged 12 to 18, to explore new ways of creating music and offer opportunities to share it. Explore new music by recent winners and highly commended entrants to the BBC Proms Inspire Young Composers’ Competition. This page offers a good index to winners of the 2016 competition, with interviews of the composers and their winning composition. Registering with the site via Facebook may be helpful to reach the sites full functionality. The younger composers (12-14) are tagged as (Junior Category). There are biographies of each winner. Some of the winners started writing music very young, but others started when then there ten or twelve.
The Contemporary Music Competition is for highly motivated students and highlights the music of contemporary composers. It is open to piano, string, wind, brass, and voice students of any age up to and including students attending a four-year college in California. If you live in California and might be interested, then consult the website for the application procedure. There is a category for children under 10 -- so even children from first grade can participate. Unfortunately, at this time, this page does not link to winning compositions. The 2015-16 guidelines are posted here.
One of the larger contests of young composers is sponsored the American PTA. Their program, Reflections, gives prizes for young composers from primary through high school. Each year a theme is selected from the public. The contest entries must all address the theme. While there is no central repository of musical compositions that have been award winners, you can find at least a few winning musical compositions on YouTube. PTA Reflections awards are given at both the state and national level. Here is an example of a composition by a nine-year-old winner from Connecticut and another winner from Missouri, also playing a piano composition.