are there dogs that catch mice and rats and do they do it better then cats

 Read up on terriers!

Jack/PArson Russel Terriers come to mind.  My parents' oldest one was the best darn mouser I'd ever seen.  That said, our Siberian/GSD mix was pretty good at killing small rodents, too.

rat terriers, jack russel terriers- can take out rats far better than cats, not so sure about mice, cats might be better there

I think that a terrier would be the best domestic canine mouse hunter and probably better than many cats. My old lab mix caught a mouse in a barn once, so I guess that shows that any dog breed can catch mice.

Wolves and coyotes catch and eat mice as a part of their regular diet so they must be pretty good at it. Are they better than one of the wild cat species? I'm not sure there's a way to tell...

If Gypsy had her way, there would be NO tree rats in our town.  The squirrel population here is BIG and I have had her go UP IN a tree at the end of her leash after one of the little beggars. She would be happy to kill them for me! Sooo, ad an Am Staff (a terrier), she fits the bill!

German pinschers were originally bred to be "ratters." 

 BUGSY!!!

huskymom

 BUGSY!!!

ROFL!!!!

On a serious note, terriers are probably best at rodents that go to ground, but in general my money is on a cat with good instincts. (Some domestic cats are lazeeeeee, but I have 3 feral rescues and they take pest control seriously.)
 

As a kid I had a rat terrier; he caught many mice in our house.  Currently I have a cairn terrier.  We get rats due to our bird feeders.  He hunts them down and gets them.  Most terriers were bred to kill vermin.

Chinese Cresteds were also historically used as ratters on ships.

My little terrier keeps us rodent free, and would like to keep the pet stores rodent free, as well 

My black lab mix is a really good mouser. But she can't get into the ceiling.. Lol. 

My golden retriever likes to catch mice, actually, he'll just get down and sniff them, and they actually have heart attacks! As sad as it sounds, it's kinda funny

any dog can be a ratter.. proof is in the working dog forum i go to. but some favourites of theirs would be the patterdale and the plummer - but the plummer is a bit of a hot topic depending who you ask. some are lovers, others are haters and it all has to do with the man that "made" them"