There is no reason to ever buy a dog or cat! Many groups actively work to provide homes for purebreds and purebred mixes. Please see the list of groups nationwide below.

CATS:
Abyssinian National Abyssinian Cat Rescue
Scope: National (US)
Website: http://abys.net/national-abyssinian-rescue-and-adoption/
National Aby Rescue on Yahoo! Groups Southern California Abyssinian Rescue, Inc. (S.C.A.R.)
Scope: Southern California area and the CFA Southwest Region
Website: Southern California Abyssinian Rescue, Inc.
Bengal Bengal cat rescue groups
Burmese National Alliance of Burmese Breeders Rescue Center
Scope: National US and Canada
Website: National Alliance of Burmese Breeders Rescue Center
Chartreux Chartreux Rescue
Scope: National (US)
Website: (Chartreux Rescue Currently offline) Chartreux cats at Petfinder
Chausie Chausie Rescue
Scope: National (US)
Website: Chausie Rescue
Twitter:@ChausieRescue
Cornish Rex Cornish Rex Friends
Scope: National (US)
Website: Cornish Rex Friends Cornish Rexcues
Scope: Southwest US
Website: Cornish Rexcues
Devon Rex Devon Rex Rescue League
Scope: National (US)
Website: Devon Rex Breed Club Rescue
Havana Brown I have not been able to find any active rescue websites devoted to the Havana Brown. There are, however, some listings here: Havana Brown cats at Petfinder
Japanese Bobtail I have not been able to find any reliable active rescue websites devoted to the Japanese Bobtail. See Manx below, as well as some listings here: Japanese Bobtail cats at Petfinder Also see the Cat-Chi Cats cattery FAQ page.
LaPerm LaPerm Fanciers International
Scope: National US
Website: LaPerm Fanciers International
Maine Coon Maine Coon cat rescue groups
Manx Tailless Cat Rescue
Scope: North Carolina
Website: Tailless Cat Rescue
Also accepts Japanese Bobtail, American Bobtails and Pixie-bobs.
Multi-breed Rescues handling multiple breeds: Multi-breed rescue
Norwegian Forest Cat
Norwegian Forest Cat Fanciers Association
Scope: National (US)
Website: Norwegian Forest Cat Fanciers Association Norwegian Forest Cat listings at Petfinder
Ocicat Unfortunately, the link for Ocicat rescue is no longer live. If you know of a rescue that handles these cats, let us know!
Oriental Slinky Cats, Inc.
Scope: Northeast US
Website: Slinky Cats, Inc.
Persian And Himalayan Persian cat rescue groups (including Himalayan)
Ragdoll Ragdoll cat rescue groups
Russian Blue Russian Blue Breed Support Adoption And Rehome
Scope: National (US)
Website: Russian Blue Breed Support Adoption And Rehome
Savannah Savannah Cat Rescue
Scope: National (US)
Website: Savannah Cat Rescue
Savannah rescue on Yahoo! Groups
Scottish Fold Scottish Fold Rescue And Adoption Network
Scope: National (US)
Website: Scottish Fold Rescue And Adoption Network
Siamese
Siamese cat rescue groups
Siberian Siberian Cat Rescue
Scope: National (US)
Website: Siberian Cat Rescue
Snowshoe Snowshoe Cat Rescue Group
Scope: National (US)
Website: Snowshoe Cat Rescue Group
Sphynx Sphynx Cat Rescue
Scope: National (US)
Website: Sphynx Lair
Tonkinese Tonkinese Cat Rescue
Scope: National (US)
Website: Tonkinese Breed Association
Turkish Van Classic Turkish Van Cat Club
Scope: UK
Website: Turkish Van Cat Club Welfare and Rescue