Dogs
Rabies is required almost everywhere. Bordetella is required by many salons and recommended by most of the rest, because kennel cough spreads through the air between dogs waiting their turn. Some also want DHPP.
Cats
Rabies and FVRCP are the usual two. Cat-only salons sometimes ask about FeLV status for cats that go outdoors.
When they check
At booking, and again at drop-off. A salon that finds a lapsed rabies at the counter has lost an hour it cannot fill, so most would rather ask early. Have the dates to hand when you call.
Proof
The certificate or the clinic’s record with dates. With PawPass you send the share link when you book and show the QR code at the door; both open a read-only page with every vaccination and its date. See PawPass at the groomer.
Sources
Rules and guidelines change, and businesses set their own policies. Confirm with your vet and with whoever is asking.