Looking on the internet deeply has found these results:

the dog owner is the most popular phrase on the web.

the dog's owner 1370,000 results on the web	the dog owner 18,650,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
This. Find the dog's owner and give it back. The dog's barking at air. Hitchcock will announce the transition through the dog's bark. The dog's owner said no. The dog's owner? The dog's in custody, at a secret location nowadays, because owners started mounting these mad rescue bids. I think the dog's still growling. When it got a little red, I called the dog's owners, but they were out of town with the dog.	She's planning to kill the dogs' owners so she can keep them for herself. This. Find the dog's owner and give it back. Administration of the product in accordance with the recommended monthly dosing schedule for a period of up to 6.5 months resulted in an improvement in lameness and quality of life in dogs with established osteoarthritis, as assessed by the dogs' owners. The promoters and the dog owners. I'm not the controlling type of dog owner. When it got a little red, I called the dog's owners, but they were out of town with the dog. The dog's owner said no. The attack dog defends its owner.

the dog's owner

1370,000 results on the web

the dog owner

18,650,000 results on the web

More popular!

This. Find the dog's owner and give it back.
The dog's barking at air.
Hitchcock will announce the transition through the dog's bark.
The dog's owner said no.
The dog's owner?
The dog's in custody, at a secret location nowadays, because owners started mounting these mad rescue bids.
I think the dog's still growling.
When it got a little red, I called the dog's owners, but they were out of town with the dog.

She's planning to kill the dogs' owners so she can keep them for herself.
This. Find the dog's owner and give it back.
Administration of the product in accordance with the recommended monthly dosing schedule for a period of up to 6.5 months resulted in an improvement in lameness and quality of life in dogs with established osteoarthritis, as assessed by the dogs' owners.
The promoters and the dog owners.
I'm not the controlling type of dog owner.
When it got a little red, I called the dog's owners, but they were out of town with the dog.
The dog's owner said no.
The attack dog defends its owner.