Comps only run when there are 5 horses entered -- if they haven't run, it's because not enough people have entered their horses. You can also enter 2 of your own horses, assuming they fulfill your comps' requirements.

It may be that you've set the difficulty too high too early on so that the kitty is low. The prestige (and kitty) depend on the skills of horses entering, the frequency of running and some mystery factor that Howrse haven't divulged. If the difficulty is high, only high skilled horses will be able to see and enter it but many look for high kitty as well when competing so choose another competition. If you think that's the case, try lowering the difficulty and make sure you have 2 horses of your own that have enough skill to enter each time they open - that helps push them to the top of the page so they get seen and chosen. It can take a bit of juggling to get them running regularly and, even when you sort it, you'll get times when they just don't fill up. Good luck

brilliant, thanks both!