This could potentially be a wrongful death suit. Your main hurdles will be the qualified immunity/privilege of the county and their counter-argument that they would have put the dog to sleep regardless.
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I am very sorry this happened to you. I am assuming this is a government owned facility. In addition to the answer already provided, I would suggest that you contact everyone from your alderman to the city manager and the mayor, and maybe a couple of newspapers and television stations. At the very least, maybe you can make sure this does not happen to someone else.
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I am very sorry for the loss of your companion, Contrary to what has been said, yours is not a wrongful death action as this cause of action applies only to human beings. As a dog person and rescuer, I hate it that dogs are considered chattel (property) and therefore have few rights but that is the state of the law. You need to do as suggested and contact local authorities as well as possibly local media so this mistake is never made again. It would probably not be worth your time and money to pursue legal action against animal control. No other dog can replace your lost one but hopefully you will find room in your heart and home to take in another furkid some day soon. Again, I am very sorry for your loss.