I cannot take credit for "Terrier" as such, Ranger1. Someone said that above. But, specifically, the Jack Russell terrier and the Parson Jack Russell Terrier (and in the travelling communities, crossbreeds from them with other useful things, eg Italian Greyhound if you want a fleet dog, or Staffordshire Bull Terrier if you want a burly one) are the dog of choice for ratting in the UK. Many other crosses are less useful because of the range of snub noses in smaller dogs - and you don't want a snub nose on a ratter. Apart from muzzle length, I don't know why wire-haired terrier breeds are less used tor ratting. Maybe the smooth coat makes it easier to clean a dog after ratting and check for wounds, which need cleaning very promptly in the case of ratbites.

Until recently the KC did not recognise the Jack Russells as breeds (as it did not the Border Collie) but I think the state of play now is that the Jack Russell is recognised, but not the Parson Jack Russell.

I would not have suggested a Dachshund. Miniatures are too fragile, and Standards would I think not have the needed agility. They might have a place for going down holes, but then they would be in trouble if attacked from behind, being unable to turn because of their length. Also the long back is very prone to injury.