These are the two really big rehoming charities in the UK:
Dogs' Trust
"The health and happiness of every dog is at the heart of all our efforts, and we try to find each and every dog in our care a loving home for life. We never destroy a healthy dog."
Cats' Protection
"Rehoming is about giving them a new start, with a new family who can offer them the love and attention they deserve. We never put a healthy cat to sleep and will look after a cat until a new home is found."
Battersea Dogs' Home has a similar policy, although I know they sometimes end up being forced to put an animal down because of incurable behavior problems (e.g. a dog so savage that it's dangerous) as well as physical health problems.
I believe the RSPCA also prefers never to put an animal down, though long-term residents may be transferred to other rehoming charities to take pressure off the RSPCA kennels. I don't support the RSPCA, though, because of its hard-core political campaigns.
