I have been a rescuer for years. There ARE all breeds of dogs, both mixed and purebreds available in shelters. The group I am with rescues highly adoptable dogs from municipal animal control death rows when their time is out. I currently am fostering a purebred, with fancy pants AKC papers, 10 month old collie someone decided they did not want anymore. She is gorgeous and well behaved. We also just adopted out two purebred pom puppies, a purebred chi, a purebred german shephard, a pure parsons terrier, a maltise, a dachshund and that is just within the past few weeks that are off the top of my head. Our local shelter just a few days ago had a purebred English bulldog who was 2 years old and had a great temperment. Another had a litter of purebred dachie puppies. We always have in small breed dogs.
Being a purebred does NOT guarantee you will never end up, through no fault of the dog, in a shelter. We have had virtually very breed of purbred in our rescue at one time or another. This is of course in addition to all the wonderful mixes available.
The OP specifically stated they wanted a non-shedder. Let me tell you there is NO SUCH THING. There are dogs that are LOW shedders such as poodles, but NO shedders is a falacy. If a breeder claims NO shedder, they are not being completely honest. Also Hypo-allergenic is a crock as well. Not even sure what the heck that is. Dogs have hair, dogs get dirty....Even low-shedders that are groomed frequently cannot claim hypo-allergenic. Just marketing gimicks. And don't fall for the designer "breeds" of goldendoodles, labadoodles, yorkie poos, puggles....you are paying big bucks for "mutts" Nothing against "mutts" I think they are wonderful, but don't pay tons for a mixed breed with a fancy oodle name.
Someone back a few pages asked what happens to the puppies the petstores and backyard breeders can't sell....well I can tell you, they end up in Municipal animal controls shelters discarded when they are no longer valuable to the store or breeder. The lucky ones get adopted or placed into rescue. the unlucky ones don't make it out alive.
as others pointed out, check out Petfinder.com There are breed specific groups out there. Not all rescues are equal though, I will be honest about this. At the risk of alienating myself against other rescuers...Some make you jump through all sorts of hoops to adopt, others don't screen enough. There are many great groups that are right in the middle. You should be expected to fill out an application asking about your lifestyle and what you are looking for in a pet. This should include a vet reference. That helps me decide if the dog I am fostering is a good match both for you and the dog. I want my fosterpets to go into great homes that are prepared for them and I want my adoptive families to have a great experience. Some animals require a fenced in yard, some don't; some love kids, some don't; some are active, some couch potatos. Look for a group that fosters....that means the animals live in private homes awaiting adoption.....I can tell you all sorts of things with the animals I foster...quirks in personalites, housebroken, good on leash etc...and I also work with them to make them more adoptable if they have issues.
Most rescue group will also have the animal FULLY vetted including spay/neuter, heartworm tests, vaccines etc prior to adoption. If there are medical issues that are known, they should be very clear in their contracts. Some areas do not have access to juvenile spay/neuter, however most do. You should get copies of all vet work completed at adopton. My rescue group spay/neuters at 8 weeks and NEVER adopts out any animal that is not fixed. And BTW...we have been doing Juvenile spay/neuter for years and have never had a problem...in fact the puppies and kittens seem to do much better than the adults with their recovery.