If my father died, mother is still living…and a sister is claiming she has power of attorney, how do I make her PROVE she IS in fact POA?

Considering the lifestyles of your parents... what? What about their lifestyles, and what's lifestyle got to do with whether or not your sister does in fact have power of attorney for your mother?

The POA doesn't matter until there are decisions to be made. At which point, if she is to make the decision on your mother's behalf regarding, say, a facility or a bank or a realtor, your sister will be obliged to show documentation and won't be able to act without it. But are there decisions to be made now which other family members wish to make on your mother's behalf? Or are you only suspicious about your sister's management of the bank account?

My guess is that she is unwilling to show any documentation because she doesn't have any. My guess is that she is using the term POA very loosely, and basing her claim to have POA on her being named as a joint bank account holder with your mother and quite possibly believing that it works like this. It is in fact, of course, putting the cart before the horse, but you'd be astonished how many people say "oh, uh, yes, I'm POA" when actually they mean they are next of kin, chief grocery shopper, geographically closest relative, in the room when the forms are being filled in, or whatever.

I would at this point simply tell sister that you are certain she DOES have POA and is handling all of Mom's bills, documentation she is required to do meticulously, but unless she can show you that you will have to see an attorney, take her to court to produce documentation and POA or simply show you her POA document. That will make it clear to her there could be a big mess if she doesn't produce the documentation done by her Mom WHEN MOM WAS NOT DEMENTED.
If she doesn't wish to do this I would get an elder law attorney for the next move.

If it turns out she isn't the PoA, then what? Is she suspected of theft or fraud? You don't really give much detail. If I were in your situation and you actually have some suspicion or proof of theft/fraud, then I'd consult with an elder law attorney (but do not tell sister this) to see what is advised. Laws can differ by state, so that's why you have to talk to a professional for your mother's state.

FYI even if she is the PoA it usually isn't in effect until there is one or more medical diagnosis of incapacity (depending on what is stipulated in the PoA document).

I am assuming that your Mother is incapable of speaking to you about this? Because the simple answer would be to ask your Mom if your Sis has POA.
If your Mom is suffering from dementia, what concerns do you have about POA? Do you suspect your Sister of fraudulent behavior? Of embezzling funds? Of doing an improper job?
When your Father died was your Sister ALSO the executor of his will and estate?
If you have reasons for concern (you haven't told us any), then do open a case with APS for wellness check on your Mom, let them know you have concerns, and let them know what those concerns ARE. Then provide any and all proof you are able to APS. While they will not come back to you with ANSWERS they will come back to the reporter to say they either did or did not find reasons for concern.

You may want to consult with an attorney specializing in elder law, but if your sister has been caring for your mother (or overseeing her care) and is on her bank account, then she has gone a long way to establish herself as your mother's caregiver. Do you have any reason to think that she is not doing a good job?

she can have a joint bank acct with your mother but that is not a POA.

What does she claim POA to do? Just the bank account or does she also make medical or financial decisions? Wouldn't the dr or financial establishment need to see a POA before they let her make a decision?

A POA can be revoked and a new one put into place. If you sister is on the bank account the she is an owner too. You mother can write a check. Things can be put into a living trust. You may need to spend a little money and speak to an attorney.

Good luck.

This issue comes up fairly often and I have enough curiosity about the answer that I did some basic searching online but nothing really comes up.

My money is on that the PoA doesn't have to prove it to any individual (as opposed to an entity like a bank or hospital, etc), unless there's evidence of financial fraud/abuse and they've been accused... but find it hard to believe this could really be the answer. I've emailed our family's elder law attorney and am waiting for his answer. I will post it when he responds.

I had to get a lawyer. My sibling was on mom's accts. Told me I had no business asking questions, or knowing anything. Told me my parents entrusted her with poa/wills, bc I would spend their money and not pay their bills. Blah blah blah. My mother told me that wasn't true, She never had dementia. Tried to scare my mom about me as well. Tried to tell ger I didnt care. Couldnt eait for her to die. Not at all. Mom told me the lawyer said it usually goes to oldest child. She could have changed it, but didnt.
If you have ANY feelings something isn't right, no answers given, fishy things going on, get a didn't. Immediately!!!
My lawyer told me I should have never waited. Sibling up to all sorts of shenanigans. He said I waited too long. I believed her and had no proof of anything. Lied to me over and over.
You are entitled to see a list of assets that are given to the court when a person dies. I never got that. Of course your mom hasn't passed. You are also entitled to answers.
Took my sibling to court and found out all sorts of things. I won't go into but it was bad. Very bad.
So get a lawyer and she will have to show proof she is poa. And can get an accounting of things.
My mistake was I waited. Don't wait.

Is mom at home or at a nursing home? They make you sign a stack of papers there. They would know. Or if mom went to hospital for anything. You could ask them.