Unfortunately, this question does not have an easy legal answer.  Unlike cruelty laws or impound laws, no state appears to directly address the issue of lost pets in its statutory code.  Indeed, while many states define dogs and cats as the personal property of their owners by statute, these states exclude domestic animals from their lost property statutes.  This is ironic considering the understood value we place upon companion animals in our society and the level of regulation states apply to animals.

The common law (the law that developed as a result of court decisions) generally holds that a finder of lost property has rights superior to anyone else in the property, except the true owner.  Dogs and other companion animals are considered the personal property of their owners.  Thus, the short legal answer to the question above provides that if a rightful owner finds his or her dog, he or she then can assert ownership.  The reality that a court may consider other factors, such as how long the person who finds the dog has cared for it, the efforts that have been made by the original owner, and the relative "value" each party has invested in the dog in terms of veterinary or other care. 

One important distinction must be made first when considering this legal question; that is, what is the status of the "finder?"   

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Interesting laws concerning lost pets.

Hawaii-
http://www.animallaw.info/statutes/stushist143_1.htm#III
§ 143-10 Stray dogs.
Every person who takes into the person's possession any stray dog shall immediately notify the animal control officer and release the dog to the animal control officer upon demand. If there is attached to the dog a license tag for the then current year, the animal control officer shall notify in writing the person to whom the license was issued, at the address given in the license certificate, and, upon demand made within twenty-four hours thereafter and without charge, shall release the dog to the person. If no person lawfully entitled to the dog, within nine days after the date of giving the notice, shall claim the dog, the dog may be sold or destroyed by the animal control officer in the manner provided in section 143-8 for unlicensed dogs.

Maine-
http://www.animallaw.info/statutes/stusmest3401_4162.htm#s3913
Persons finding stray dogs. A person finding a stray dog and taking control of that dog shall take that dog to its owner if known or, if the owner is not known, to the animal shelter designated by the municipality in which the dog was found.

Two things become apparent from these two statutes.  First, both states require that the dog is either returned to the owner or given to an animal control officer or shelter.  Second, these laws imply that the finder cannot retain the dog and must turn it over to municipal officials. 

This issue is further muddied by the fact that twenty states have enacted lost property statutes that set out a procedure when lost property has been found.  These statutes set out a series of steps a finder must follow when coming upon lost property.

Despite the fact dogs are considered personal property and no other statutes concern pets as lost property, these provisions may not apply to companion animals.  Of the approximately twenty states and District of Columbia that have lost property sections, two specifically exclude domestic animals from their application (New York and South Dakota).  Those states that do not explicitly exclude animals from their lost property statutes employ a statutory procedure for finders of lost property. 

Michigan-
http://www.animallaw.info/statutes/stusmist287_261_395.htm#s302
287.305 Lost dog; finder entitled to fee for keeping.
Sec. 5. Any person finding a dog registered under the provisions of this act shall be entitled to the sum of 25 cents per day for boarding such dog, such board to be paid by the owner. The commissioner of agriculture shall furnish such finder with the name and address of the owner, upon request.

287.308 Stealing or holding dog in possession; penalty.
Sec. 8. Any person who shall steal or take without the consent of the owner and without lawful authority, any dog registered under the provisions of this act, or any person excepting dog wardens who shall harbor or hold in his possession any stray dog of which he is not the owner and does not report such possession to the sheriff of the county or the police department of the city in which he is holding such dog within 48 hours after such person came in possession of said dog, where the value of such dog shall not be in excess of $100.00, shall be guilty of a misdemeanor, and where the value of such dog shall be in excess of $100.00, shall be guilty of a misdemeanor and upon conviction thereof shall be fined not less than $50.00 nor more than $500.00, or imprisoned in the county jail for not more than 1 year, or both such fine and imprisonment in the discretion of the court.

New York-
http://www.animallaw.info/statutes/stusnyagmktslaw106_126.htm#s113
§ 113. Change of ownership; lost or stolen dog
2. If any dog which has been assigned an official identification number is lost or stolen, the owner of record shall, within ten days of the discovery of such loss or theft file with the commissioner a written report of such loss or theft. In the case of a loss or theft, the owner of record of any such dog shall not be liable for any violation of this article committed after such report is filed.

North Carolina

http://www.animallaw.info/statutes/statestatutes/stusncset.htm

NC ST § 130A-195  

This North Carolina statute provides that when quarantine has been declared and dogs and cats continue to run uncontrolled in the area, any peace officer or Animal Control Officer shall have the right, after reasonable effort has been made to apprehend the animals, to destroy the uncontrolled dogs and cats and properly dispose of their bodies.  

NC ST § 160A-186  

This North Carolina statute provides that a city may by ordinance regulate, restrict, or prohibit the keeping, running, or going at large of any domestic animals, including dogs and cats. The ordinance may provide that animals allowed to run at large in violation of the ordinance may be seized and sold or destroyed after reasonable efforts to notify their owner.