To view a purebred horse's stars, you will need to register them in their appropriate Breed Registry which can be found under Town. Recognized crossbreeds can also be registered into their appropriate breed's Club, which is also found under Town. The wiki page for registries has additional information about costs and benefits for joining and registering horses.

Adding onto this. How does one go about Grades? My grade Thoroughbreds get accepted into the one club I am in ran by Maki, The Thoroughbred Association and my Marwaris, are in my own club I have for them. I have some grade Percherons, Fjords and then Friesians, but I am not sure where exactly to put them at as trying to search clubs is a bit prickly at the moment.

Grade horses are unable to be entered into Breed Registries. As per the wiki, they can only be entered into clubs and will receive purple, not yellow/gold stars.

Mine received grey stars or rather silver.

Crossbreeds is purple though. So like registered ones; Warlanders and such.

But thank you.

I cant even register my grades in clubs haha