Pet overpopulation means that there are too many dogs and cats and not enough good homes for them all. It is hard, but not impossible, to find a good home for a dog or cat that you find. In the meantime, try to find the owner and/or temporarily provide a safe place for the dog to stay at your house while you look for options. Be sure to keep the dog crated or away from your animals and children while you evaluate this new pet. Most stray dogs are NOT aggressive, just lost or dumped by an irresponsible owner who cannot care for them.
If you must take the dog or cat to a shelter that euthanizes unwanted animals, know that there is a good chance that a stray will not be claimed by its owner in this facility, despite the fact that shelter personnel will advise you to bring it to them. Be proactive in trying to find the owner but don’t put the dog back into a dangerous situation. And beware of advertising this found dog in a way that someone might claim that it is their dog, to get a free dog. We have had this happen, especially with pit bulls. Be smart and have the caller identify the dog with information that you have not supplied.
Do not dump your found dog or cat. While it may put you in a difficult position to take on an added responsibility, finish the job and help the dog or cat by making sure it is out of harm’s way. Putting it back on the streets is the worst possible answer. Use the resources that you have and by all means, help this innocent victim.