My son adopted a pit bull that possibly had some lab in it. It was a rescue, but not from a shelter. My son worked at a large chain pet store for a short time after high school, and someone brought the dog in, and left without it. They just abandoned it. No collar, no microchip, no tags, no info. At the end of the day, my son and his roommate thought the dog seemed nice, and brought it home.
He wanted to keep it at his apartment, but I insisted he bring the dog to a vet for a checkup. We knew nothing about the dog. We told the vet that adoption was being considered, but not if the vet suspected the dog was sick, etc. But what we found out was that the vet could also tell a lot about the dog's past, and it's temperament. The vet interacted with the dog for quite some time, and gave it a couple of basic shots. She based her opinion on how the dog reacted to her, how the dog accepted its shots, how the dog responded to her voice and actions. Her opinion was that the dog had somehow been treated extremely well. We'll never know why it was abandoned. But the vet said she (and other people who are familiar with dogs, or dog trainers) can tell how a dog has been treated by the way they act and respond. It was all new to me, but it made sense. It has to do with the way the dog behaves, how it looks at the person, etc. The vet said that she would not have any qualms about adopting that dog herself.
So my son took the dog and had it for several years until he gave it to a friend who treated it equally well (my son travels too much for work). The dog was amazingly good - friendly, calm, and obedient.
So you might ask a dog trainer or vet to evaluate a dog and get their opinion. Tell them you're considering adoption, and what kind of home you have (little kids, noise, other pets, space available, schedules) and ask for an evaluation whether this dog would be a good mix.
The other thing to consider is, a lot of towns and homeowner's associations and apartment complexes ban these breeds. So do some campgrounds and hotels and other places. So you might explore where you live and where you travel, and make sure pit bulls aren't excluded.