If you have it selected as one of your VIP perks then you will see a little gear icon on the competition tab. Click it and itll glow white and then itll auto register you in any of the comps you click for the one your horse is most likely to win.

The description for the perk states: "If your horse has less than 20 victories, it enters comps where the chance of winning are the highest. If your horse has more than 20 victories, it is registered to the most profitable comps."

While this is technically true, it depends on how you have the comp page sorted. If your horse has less than 20 wins and have the page sorted by lowest or highest difficulty, that criteria overrides the "best chance of winning" criteria.