you can find ANY breed of cat/dog in shelters/rescue groups, though they may not be young puppies. In my experience, and I volunteer at etobicoke humane society and have fostered for Mississauga humane society, the animals just need TLC.
Almost all of the cats I've seen (and its hundreds) are VERY friendly and nice. The dogs may have a few bad habits though from previous owners which is unfortunate (not toilet trained, etc.). I've worked with great dogs though, even the medium sized ones who at first I thought were loud yapmouths, when they get used to you, just adore you.
You can offer to foster dogs (you can say yes or no to the dogs they ask you) and they will pay for all the food/etc. If you end up liking the dog then just adopt him (if nobody else does). Of course these are usually older dogs, but for me I don't mind because I love large dogs.