I had this come up last year, caused me to call my attorney and do quite a bit of reading. First thing I learned, there are "Service Animals" and there are "Emotional Support Animals." The rules are different for the two distinctions. And while I've found quite a few people have anything from a pit bull to a duck for emotional support, I've haven't had this happen for actual service animals yet.
Here in Colorado, you can't register to vote if you don't own a dog. Just kidding. :-) But seems like EVERYONE has one, this comes up every time I list a place for rent. I'm not an attorney, this is not legal advice! But I found that while you have to make reasonable accommodation for either of these distinctions, changing insurance companies is deemed to be unreasonable. I was quite pleased to discover that my insurer forbids certain breeds, including pit bulls. In my written rental requirements, I state no animals not covered by my insurance. These requirements are only given out at a showing after an applicant has passed my phone screening, or if they request it sooner, but I don't offer a copy immediately. I've also changed my phone screening question to "Do you have any animals?" instead of "Do you have any pets?", this lets me know what I'm in for right away.
It's odd, but you can ask what a service animal has been trained to do with regards to a disability, you can't ask ANYTHING about an emotional support animal except to confirm that this is the animal's role.
@Brian Simmons I really like how you've handled this with regard to verifying certification of the animal. I have managed not to have it go this far yet, but will use your method if it ever does.