Tootsie, our Blue Heeler mix, is great for going after mice. She doesn't eat them, but she dos seem to know that they do not belong in the house, and when she would see a mouse, she would go right after it.
We don't have any mice here, but before, we lived in a run-down trailer house out in the country; that place had more holes in the floor in the closets than you could count, so the mice would easily come right in. We had some food stored on the back porch, so I had to be careful not to put anything out there that the mice might get into. Tootsie was a great patrol dog, but she could not be everywhere at once.
Now, we have an opossum that comes up by the back door at night looking to see if there are any scraps of leftovers out there for him to eat. Tootsie sleeps just inside the back door, which is mostly glass, so she has a complete hizzy fit when she see that possum out back.