Disclaimer This document is based on several years of discussing allergies with others on the Internet. I am not a doctor nor a veterinarian and my allergies are confined to cats and some long haired breeds of dogs, easily controlled with frequent hand-washing.
If you're looking for something the other way around, eg, allergies in dogs, try the Canine Allergies FAQ. I don't have anything specific for other animals, sorry.
Truth is, there is no hard and fast answer. There is no group of dog or cat breeds one can point to and say "these are hypoallergenic". People have differing levels of allergy and allergies to different things -- sometimes it's the hair, sometimes the saliva -- although most often it's the dander. The only way for an individual to be certain he or she is not allergic to a particular breed is to spend some time with individuals of that breed and see how he or she reacts to the animal. Keep in mind that for virtually any breed of dog or cat, there is a person allergic to it.
You should also be very careful in your quest for a pet. You must be honest with yourself: if you later must give up your pet, it is much worse for your pet than it is for you. It is a sad fact of life that many pets given up by their owners, especially once fully grown, are put down at shelters. So, do not focus so much on your own desire for a pet, but rather consider the potential pet's welfare first.
Small dogs are probably "hypoallergenic" because there is less of them to shed or otherwise distribute allergens Any of the toy breeds warrants consideration under this category.
Dogs with little or no hair may be good candidates for some, because the amount of hair shed is so much less. Breeds in this category include: whippets, Mexican Hairless, Chinese Crested, Boxers, greyhounds, Basenjis, shorthaired Chihuahuas, etc. However, for some sufferers, they exacerbate the problem, because the dandruff and saliva is more accessible, so watch out!
Dogs with "human like" hair usually have long silky hair that sheds minimally. Many people seem to have limited reactions to this type of coat. Breeds falling into this category include Yorkshire Terriers, Maltese, Bichon Frise, Havanese, Silky and Skye Terriers.
Dogs with nonshedding coats are the most commonly recommended breeds for allergy sufferers. However, it is important to note that there are some people who can't tolerate these dogs, either! When a dog's coat does not shed, there are less allergens released in the air. This is provided that the coat is kept clean and clipped; a Poodle with a corded coat (think of dreadlocks) could drag in as much dust and allergens as a furiously shedding Samoyed! Breeds that do not shed include all three sizes of Poodles, the Komondor, and the Puli.
Puppies often have different coats than dogs do, it's important to check that you are not allergic to either puppies or adults of the breeds you are considering. For example puppy Standard Schnauzers may trigger allergies while adults often do not. Conversely, Golden Retriever puppies may not trigger attacks whereas an adult probably will.
Finally, don't forget to balance your allergy needs against the kind of dog you want. If you are a retriever lover, you may not be happy with any of the terriers, even if the terriers are what you are not allergic too. Don't overlook personality compatibility in your quest for a hypoallergenic dog!
If you have ever been hospitalized for an allergic episode or an asthma attack, you are likely risking your life in trying to get a dog or a cat, and in this case I recommend that you simply not get one. It's not fair to either you or the dog.
Some types of allergies worsen over time. You might select a dog that you later develop allergies to in any case. This is why it's very important for you to find a good allergist who will educate you about your particular form of allergy and help you to manage it.
Do keep in mind that some people do have serious enough allergies that they just should not have a pet and a good allergist will tell you that, too, after checking it out thoroughly for your case.
I would contact different breeders of the breeds you are potentially interested and explain that you're not sure about allergies to that breed and is it possible for you to spend some time with their pets and find out? Most responsible breeders will be happy to help you find this sort of thing out before you commit to getting one of their puppies or kittens.
The magazine New Woman (October 1992) has an interesting article about a cat-allergy vaccine. Catvax is being developed by the Immulogic Pharmaceutical Corporation (I.P.C.) in Cambridge, Massachusetts, and is now being tested on humans at Johns Hopkins University. Tests on animals indicate that Catvax is different from traditional cat-allergy shots in two ways. First, unlike conventional allergy therapy, which involves biweekly or weekly injections for up to a year, the vaccine may be able to completely prevent allergic reactions after just a few injections. Second, studies suggest that the vaccine will not produce allergic side effects, such as asthma, that traditional shots often do. I.P.C. hopes to complete its human studies and have the vaccine on the market by 1996 or 1997.
There is an informative article "When Humans Have Allergies: Ways to Tolerate Cat Allergies," in Cats Magazine, April 1992.
The August 1992 issue of Cat Fancy contains an informative article; the September 1992 issue has a survey of people's experiences with allergies and what worked for them.