I found my girl Sophie on Petfinder. I would recommend it as the first place to go when looking for a companion animal. It has easy search options. Just key in your location, what type of creature you're looking for, breed, gender, age, size, optional questions are does your home have cats/dogs/children.
Then the site filters through and presents you with a photo album of animals who might fit the profile you've created. It's wise to record maybe four possibilities. Also remember that some shelters won't have an animal filed by name. Definitely have any case numbers as well.
I feel it's also good to have several options so you can make a better choice. You only know if a creature is a good fit for you by meeting them. Not just from an online description. An example, I was taken by a photo of a male dog at a local shelter. When I actually met him he was much larger than his picture suggested.
They also hadn't mentioned his tendency to hump humans. He was so wound up he ripped my jacket as I tried to snap on his leash. Sadly I had the warden take him back to his kennel. The story ended happy though... The pup in the next cage turned out to be a winner and has been with us for four years now.