I volunteer at an animal shelter with the kitties, so I spend lots of time with them, know them all very well, and know how they react with people who walk in compared to us volunteers.
So a couple of tips:
1. Find a shelter with volunteers who know the cats. Listen to the volunteers. If there's a cat you really like and they say "this cat doesn't like other cats/children/noisy households" etc and you have one of those things, please really rethink whether cat that might be right for you.
2. Tell the volunteer your living situation and what sort of cat you're looking for. They should be able to tell which cats would suit you. If you don't like any of those cats, then maybe try other shelters, or come back another time. You don't HAVE to get a cat the first time you go to a shelter. It pays to look around and be patient. This is your furbaby for life.
3. If you can, go during the week when it's a little more quiet. The weekend volunteers at my shelter don't know the cats as well because it's so busy on weekends, they don't get time to get to know them. During the week, the volunteers get more one on one time with the kitties so know them a lot better. Also, if you go during the week, you're more likely to be able to spend quality time with cats you're interested in. On weekends, there's often too many people to do that. As well as that, the cats get stressed with so many people coming through, so you won't see their best side.
4. Spend lots of time with the kitty you're interested in. Talk quietly when you're spending time with them, don't move too fast, and don't expect them to want to play or want to cuddle immediately. Just sit still for 5 minutes and watch them and let them get to know you. Like I said, no matter how well they're cared for, it's a stressful situation, so give them time to settle down with you and show their real personality. If there's a play area you can sit with them in, make sure you don't have the entire family in there - maximum of 2 people is good.
5. If you have small children, and want to adopt a kitten, prepare yourself for the fact that the shelter may not let you adopt a kitten. At our shelter we have a rule that families with children under 6 can't adopt kittens under 4 months of age. They're very fragile, and can't run and hide as easily. We had a number of kittens come back with broken legs one summer a couple of years ago which got this rule enforced. Your child doesn't NEED a little kitten - wait until they're older and can understand the responsibilites of having a little kitten, and enjoy it more.
6. Find out which kitties have been there the longest and ask about them. We have a gorgeous kitty at our shelter who's been there 5 months now! He's a shy boy with a damaged eye that looks horrible, because a cat beat him up in his previous home. When he gets to know you though, he's the sweetest most lovable kitty. Another girl and I have spent lots of time with him, and now he loves us. When other volunteers have him in the play cages, he hides and won't come out. With us, he never hides. He cuddles, he plays and has a great time. This is the REAL Inky. When he first came to the shelter, he wouldn't even come out of this cage to play in the play room. Now he jumps down from his cage on his own and runs into the play room as soon as you open the cage door. He has improved sooo much since he's been at the shelter. Unfortunately the public will only see this side of him when the other girl and I are around. Even if these long term kitties aren't a kitty you could take, maybe you know someone who could give a home to a shy kitty who needs a quiet, indoor home.
Mmmmm I think that's about it for now...
Good luck in the search!