Crossbreeds cannot show in halter.  This means until they are 2 years old they cannot show, but once they age up then they can do discipline shows.

Edit: I should add this is only true for generic crossbreeds, if your crossbreed has been made official in the game via completed campaigns then they can Halter show through their club.

I am going to add on to what Meriweather said

Crossbreeds without an active campaign or without a registry cannot halter show as foals or adults (Campaigns have halter shows whether in progress or completed)

All horses whether a crossbreed or not, need to have the related working path selected and if entering Disc shows or Careers also needs an option chosen in the related drop down boxes.   After making any changes you need to make sure to save with the green check at the top.  To edit these settings for a specific horse head to their page and click the edit botton (last button, looks like a square with a pencil)
If entering Careers - your horse also needs to be personality tested at the vet if they did not have foal training completed 

if you mean this foal, there are no clubs for a 50% mustang/50% appaloosa mix, so you'll have to wait for this foal to grow up before he's able to do discipline shows and/or career work.