Usually same day unless it needs to be nuetered or if it just arrived and hasn't been processed yet. I didn't need to show any thing but my ID.
It depends on circumstances. I put our dog on an adoption hold so I could come back the next day to introduce him to our other dog. Whatever red tape the above response is referring to, I suspect those weren't adoptions but people trying to get escaped/lost pets back, which does require a lot of proof. As it should.
Adoptions generally happen same-day and the process doesn't take very long. There are some occasions where circumstances might delay things until the next day, but that's pretty rare from what I understand.
We went, looked at the dogs, had three of the dogs (separately) interact with us in one of the private rooms, picked the one we wanted for my father and since there was no hold on her, we took Sierra home that day. The whole process took about an hour and a half.
There is a lot of red tape, every time I have been at the shelter, I witnessed folks being td they would have to return at a later date or provide some other documentation, or the animal wasn't available at that time.
Thanks! Your answer is awaiting moderation.