We spent a lot of time at our area shelters last fall when we were looking for a dog to adopt. We needed a dog that would be gentle with our older dog and good with the cats, yet still playful & good for hiking, so the search took awhile, but we found our jewel Beth after a few months of hunting.
We have two large no-kill shelters and the city animal control downtown.
The city shelter hours aren't very good but have improved - they're now open Saturdays. They select a few dogs & cats out of the thousands that come in
as adoptable, and put those in the adoption area to be kept as long as it takes to find them homes. Those animals don't have a time limit - all the other animals are held for 2 days, then euthanized. Their adoption kennel/cage area is very small, and you aren't allowed to bring in another pet to meet the one you're interested in adopting (there isn't any place to do this). Dogs and cats are housed in the same room. Fortunately, a bond election is on the table to build a larger facility, with more adoption space and space to increase the holding time to 4 days.
The older no-kill shelter in town is open very limited hours during the week, but it is open on weekends. They have a separate area for the cats, including a large area with an outdoor enclosure where cats that get along can roam freely. They have a waiting list for strays to be adopted, but in the 12 years I've lived in this city I've had many animals (I attract strays) on their waiting list and never have been able to get an animal in there. They charge equally to take in surrendered animals or strays. Their facility is nearly full of hard-to-adopt out pets with serious illnesses or behavior problems. When we looked there for a dog, it seemed like at least 70% of the dogs in their kennels were aggressive. To me that's kind of sad - it's nice that it's a no-kill shelter, but at the same time, tens of thousands of adoptable dogs and puppies are euthanized in this city every year, while these aggressive dogs live out their lives in the adoption cages.
The other shelter has a policy that they will only euthanize very ill animals that can't be treated, and viscious or aggressive animals. They moved into a much nicer facility this year. They are not open very late in the evening, but they are open on weekends. They moved into the new facility after I adopted my dog, so I haven't seen too much of the new place. They also have great education and screening programs for adopters. They accept pets on certain days by appointment - call to see if they have space, and if they do they will have you come in & they will determine if they can take the animal. I've turned in a few found strays to them. It's a lot easier to get a cat or small dog or puppy in then a large dog or puppy. They charge to take animals - the fee is determined by the animal's age.
p.s. Please don't think I'm evil for turning in strays to the shelter - our city has a HUGE animal overpopulation problem, and we have financial & physical limits on how many pets we can properly care for in our home. I believe they have a much better chance of finding a good home at the no-kill shelter than trying to survive on the street. Now that I'm in a house (instead of a tiny apartment with an irate landlord), if it's at all possible, I try to find a home for them myself. But the vet care is $$$, in addition to our own pets' care, and there are just so many.....