There are many organizations helping to rehome dogs to the right owners. Here are website addresses for some of them:
UK
Wild at Heart Foundation
Battersea
Lurcher SOS Sighthound and Lurcher Rescue
All Dogs Matter
Many Tears Animal Rescue
Dogs Trust
Royal Society for the Prevention of Cruelty to Animals (RSPCA)
Blue Cross for Pets
USA
Petfinder
American Society for the Prevention of Cruelty to Animals (ASPCA)
Animal Lighthouse Rescue
Animal Haven
Rescue Dogs Rock NYC
North Fork Animal Welfare League
New York Bully Crew
The Inner Pup