My family has adopted several dogs from shelters and rescued them from other people, too. In our experience, while we had a certain breed/size dog in mind, it was still mostly a case of seeing which dog really “called out” to us. It wasn’t always the type of dog we had in mind, but in that way it’s like dating and finding a perfect companion in someone you swore “wasn’t your type”
You could also have some rough criteria in mind, and ask which of their dogs is most in need of a new home (I.e. has spent the longest time there through no fault of their own, or, worse, is closest to being put down if it’s not a non-kill shelter).
We already knew the reputation of the shelter we adopted from, since it’s very local, but it might help you to research this, too, if you can. Are they reputable? Most good shelters would, for example, do things like make an effort to match you with the right animal in the first place, take them back if it didn’t work out, and do home checks/a few post adoption follow ups.