If your pet is older, less adoptable, but is spayed or neutered, you "MIGHT" be able to find it a good home. Ask a small fee, and be willing to do home checks. Do not allow a dog to go to a home without a fenced yard. Make sure any prospective owners either own their home or have landlord permission. Are their other pets in good health? Do not be in too much of a hurry to get rid of your pet that you place it in a questionable home. Do not give an animal to anyone who will not give you their name and phone number.
The Truth
The truth is that unless your pet is under two years of age, or is of an unusual breed, it probably wont have much chance at finding a new owner. Small non-shedding dogs are in demand, but an untrained, older, active, big dog will not have much chance, especially if it is not spayed or neutered. Just because you love it (or maybe you don't) does not mean any one else will love it. Accept the truth from a shelter if they tell you that your dog (or cat) is not adoptable. It will be less stressful to these animals for you to take them straight to the vet for euthanasia.
The reality is that more animals are born every year than there are homes for.
Stray Animals
If you have a problem with animals in your yard and are trying to get rid of them the only thing you can do (in most areas according to law) is to take them to the pound, SPCA, animal shelter, etc, whomever deals with strays in your area. This is too allow an owner a chance to find them, or for them to be put for adoption to find a new owner, or to be euthanized humanely. Picking up strays and dumping them in other parts of your city or country is generally considered Theft and Animal Abandonment.
In most areas if your neighbours pet makes a nuisance of itself coming onto your property you can also take it to the pound or shelter.
Always think carefully before getting rid of a pet, once you have signed it over to somebody else you may not be able to get it back. To avoid being upset, it is best to plan carefully BEFORE getting any pet.
We adopted 2 cats recently from the RSPCA that also had been dumped in a remote village. They were found in a cardboard box.
What drives people to dump their pets is beyond us! It is unnecessary & extremely cruel.