Interesting article here about discovery of a dog 33,000 years old. That's a long time ago. The article talks a lot about "morphology" (size and shape) and DNA analysis. The author argues that this isn't really the beginning of domestication of dogs, the "morphology" is pretty much pure wolf and the DNA matching is more wolf than dog. He speculates that this specimen represents an early attempt at domestication that didn't work out, or the harshness of the last ice age which started maybe 20,000 years ago, aborted the domestication.
In actual fact, the difference between domestic dogs and wolves is psychological, more than anything else. Dogs have a much better attitude about humans than wolves do. Dogs will accept petting, food, affection, and even obey orders. Wolves, not so much. Modern German Shepherds and Huskies look a lot like wolves but aren't. It's not clear to me that this psychological difference would make much difference in the DNA or in the shape of bones.
It's a good bet that wolves were domesticates when some human children came upon orphan wolf pups in the wild. When young, pups are cute, humans are attracted to cute, and surely the children carried the pup[s] home and made pets of them. Some wolf pups so adopted must have carried the genes needed to bond with humans. Probably other orphan pups lacked these genes and ran away when they grew old enough or were driven off when they did something wolflike such as threatening small children. Some how, an adopted wolf cub who hung with the humans must have found a mate somewhere, and gave birth to a litter of pups while living with humans. It would only take a couple of dog generations to establish a domestic strain of dogs that were breeding while living with humans. Dogs are pretty useful in the hunt and in guarding the camp. The humans would have taken to them. Despite the harshness of the coming ice age, it's hard to see that breaking up the domestication of dogs once begun.