They can be anywhere in town, at their home or wandering around town.
When I got this challenge I already new the cat with my dog and they was friends so I went to the Relationships tab and looked at the other household members the cat lives with.
After that I went to the cats home and the cat wasn't there so I waited and after a few sim hours it came home.
If you don't know the cat it may show up when you view the town map like opportunities sometimes do. Also note when I saw the cat at it's house it had a Magnify Glass icon over it to represent it's part of the myster challenge so keep a look out for that over cats in the area your dogs searching.
