Very interesting thread! I have a service dog - trained, certified and all of that.
If you are in the USA the American Disability Act is the law. No one is allowed to ask you what your disability is -- i.e. no showing pills etc, that is humiliating. They may ask you if the dog performs a necessary service for you, but that is all.
Even with the certification there are issues. Just two days ago I was at the eye doctors with my dog waiting to be seen, when one of the techs rushed up and took me into a room because someone in the waiting room complained about the dog being there. (No one mentioned the two screaming kids in the waiting room who had snotty noses

). Anyway, I told them that I would leave because I had no desire to cause a problem -- with that the doctor came out and said absolutely not, and I got to be seen with no problems. Soooo, even when everything is ligit there are issues.
My disability is fairly obvious -- I have hearing loss -- so generally I don't have a problem. That said, however, when I go out to eat with the family and am not driving and such I do not usually take her.
I carry the ADA law in my purse on a little card for those who need to be educated and am always polite about it. Now if you are out of the USA you have other problems -- Mexicana Airlines does not have to honor the American Disability Act if they are out out of the US etc. So you have to watch that.
Just a little hint. When I make a reservation for a plane I have learned not to tell them I have a service dog when I make the reservations. I wait a few days or weeks and then call and say I will be bringing my service dog. Reason??? I have been turned down for a reservation several times because I told them I had a service dog at the time of the reservations.
Hope this helps/clarifies.
BTW - training a service dog takes about $8,000 to $10,000 not counting the hotel expenses if it is not included in the price. Most people have grants to pay for the training of both dog and handler.