Dad had a massive heart attack. They are releasing him on Monday. We are arguing about power of attorney. Help me!

You said "My Mom is his wife". Does this mean you and sister share a Dad but not a mother. If this is so, then I can see why she wants POA. Otherwise, I don't see why she would trump her own mother.

As said, Dad assigns the POA which I think would be his wife. In my opinion, you should be second since you probably be the one responsible for their care not sister. Sister maybe under the wrong impression of a POA. It only comes into effect if Dad can no longer make decisions. She has no control till then. Since he is married her being POA does not make sense.

Social Workers cannot and should not get involved with family drama. Social Workers are there to help set up resources needed for Dad. A POA is a personal and legal thing. You should go to a lawyer to have papers drawn up, both financial and medical. They have to be witnessed and notarized. I would do this no other way.

Dad's choice. Unless dad is declared incompetent, he gets to choose.

It's Dad's choice, sorry to say. He can appoint the dog catcher if he wants to.  Usually the  wife is appointed, unless she is mentally incapacitated.  POA is only for the financial stuff.  The other form is the Health Care Proxy: someone who makes the medical decisions.   When the patient dies, both the POA and the HCP become null and void.  At that point the Executor of the estate takes over.   It might help to sit down with an attorney to go over all the legal forms you need.

It is good to have an alternate POA. If your dad, the principal, chooses your mom as his agent, then it’s also a good idea to have a second person appointed in case the first person can’t act as the agent. Only the principal can appoint a POA. It doesn’t matter what your mom or sister want. Dad gets to choose who he wants (trusts) to make decisions for him. There can be a POA for medical decisions and a POA for financial. It doesn’t have to be the same person but it can be. ALSO, your dad can change this POA whenever he wants to, as long as he is competent to do so.
Does sister want them to come live with her? If there is no POA generally speaking, the spouse makes the decisions if the patient can’t. If your dad isn’t able to make a decision right now, he won’t be able to appoint a POA.

It is dad's choice and decision on who will be POA. Think about bringing hospital social worker into the discussion.