If you're in NYC, you might be interested in paying a visit to the first round of the Honens International Piano Competition in New York City (other sites are Hamburg and Calgary. It's free and open to the public.
You can find more info here.
I don't know that it's one of the first to spring to mind when you think about piano competitions, but it's admirably ambitious. One thing of interest is the amount of collaborative work required. There is the usual concerto and chamber music, but also some accompanying for lieder.