Q7 from this site:
The ADA Home Page provides access to Americans with Disabilities Act (ADA) regulations for businesses and State and local governments, technical assistance materials, ADA Standards for Accessible Design, links to Federal agencies with ADA responsibilities and information, updates on new ADA requirements, streaming video, information about Department of Justice ADA settlement agreements, consent decrees, and enforcement activities and access to Freedom of Information Act (FOIA) ADA material
https://www.ada.gov/regs2010/service_animal_qa.html
In my experience (done for work) asking actual service animal owners these questions produces immediate, clear answers and is not embarrassing for either party. Asking people who are trying to sneak their pets in somewhere goes slightly differently but at that point, barring any state/local ordinances, you can enforce your business's rules.* Whatever you do, DO NOT ask them to "prove it" as that puts you in violation of federal law.
Also, further down the page (Q25-29), even service animals can be kicked out for various reasons (out of control, not housebroken, etc). Note that I've never, ever seen a true service animal out of control but there are methods of recourse for inconsiderate and disruptive pet owners beyond throwing up your hands.
*My job and position are somewhat unique in that people are naturally less inclined to argue with me regardless of whether their animal is a service one or not. I accept that my interactions may be more pleasant than those of a restaurateur. That said, I think people with legit service animals are more than happy to answer the ADA questions because they're a bit like Celiac sufferers in a world of "gluten intolerance"--tired of idiots giving them a bad rap.