einzwitterion:

I have ordered wallet size cards that summarize ADA (Americans with Disabilities Act) laws that apply to service animals. There are 50 in a pack, so if anyone wants some, let me know and I will send you a few! I plan to keep one or two with me when out and about so I have the resources to give anyone who is discriminating against service animals.

“The only two permissible questions are whether the animal is required
because of a disability and what work or task the animal has been
trained to perform. Further, under these same regulations, those
inquiries cannot be made if it is readily apparent that an animal is
trained to do work or perform tasks for an individual with a disability
(for example, it is usually pretty obvious when an individual is using a
seeing-eye dog).” [Source: summary of ADA laws on service dogs from the American Bar Association]

You can’t ask to see paperwork or if the animal is registered as a service animal. There is no government-sanctioned registry of service animals, so a registered service animal is not really something that legally exists. Service animals are not required to be trained by a professional organization, they simply must be trained to perform tasks for their handler (who must have a disability).

Service animals are allowed in any public, governmental, or non-profit space provided they do not pose a threat to health and safety. For example:

• Service animals are allowed in restaurants, but they may be excluded from restaurant kitchens if the public is also not allowed in the kitchen. Members of the public generally aren’t allowed in commercial kitchens because it poses a food safety risk, likewise service animals are not.
• Service animals are allowed in the general areas of hospitals, including visiting patients on most wards. They are not allowed in sterile environments, such as operating rooms or burn units–the general public is also not allowed in these spaces.

A service animal that is not in the control of its handler (being aggressive to others, pooping indoors, etc) can legally be asked to be removed from the premises, but its handler must be allowed to stay without the animal and given the same service anyone else would have. The only species of animals that are covered as service animals under the ADA are dogs (any breed) and miniature horses up to 34″ tall and 100 lbs. Emotional support animals (ESAs) have some legal protections, but their right to access public spaces is not as broad as service animals. So you are legally permitted to prevent someone from bringing their emotional support bunny (or, even, dog, if it is an ESA and not a service dog trained to perform at least one task to help its handler with a disability) into your restaurant. ESAs are exempt from no-pet policies in housing and may not be charged pet fees while flying. Proving your animal is an ESA actually requires more paperwork than a service animal (you need a letter from a medical professional stating your need of the animal). 

tl;dr: I will send you ADA service animal law cards if you want them. And don’t fucking pet a working service animal. Don’t even talk to it. It is working and you are being annoying (and possibly endangering its handler.)