It is now a USEF rule (and penalties may be leveled) that dogs are not allowed to be loose on the grounds at a horse show.  They must be contained or on a leash.  (GR 3301.6)

As of January 16, 2006, it is a USEF rule that dogs are not permitted to be loose on horse shows grounds. They must be contained, or on a leash. And, you may not lead a dog while you are riding on your horse at any time. There will be a penalty imposed if the rule is not followed as well as the issuance of warning cards.  In addition, some shows’ insurance companies are now not allowing dogs (other than those owned by management) on their property during a horse show.  As CHSA is affiliated with USEF, we follow their rules, even though the show you are attending may not be USEF recognized.