One clear-cut benefit to working with a shelter or rescue is that they may know a pet's history and may be able to tell you if the cat or dog lived with another pet of a different species.
"Choosing a dog who's lived with a cat and vice versa is usually a good way to go," said CJ Bentley, Behaviorist at the Michigan Humane Society.
And if you have an adult cat, it's best to pair it with a similarly aged dog, since an energetic puppy may likely set a senior cat on edge.
The First Meet & Greet
Take precautions: make sure your dog is leashed when first meeting your cat. While the introductory meeting may seem to be going smoothly, it all can change in a flash if your cat makes a run for it. So it's best to make sure your cat has a clear route to escape, and your dog is restrained should his predatory/chase instincts kick in.
Never Forget What's in a Pet's Nature
Cats can injure dogs by scratching if they feel threatened. And cats can injure themselves while trying to flee.
Dogs may simply be too big or rambunctious for a cat to handle. And an unintended invitation to play can lead to injury.
"Remember too that just because an animal 'tolerates' a new family member, it doesn't mean they really enjoy the other's company," says Bentley. "And some dogs just can't relate to cats as housemates."
Best Left Alone
Until you're certain that the cat and dog have accepted each other, it's best to keep the two separated when left alone.