I think for most of the history of cat domestication there was no attempt to selectively breed cats. The cats went out into the night and made their own arrangements. Dog breeding has been controlled for a much longer time, people have been trying to breed dogs for particular jobs for thousands of years, cats maybe for the last hundred or two.
The first link that Rhythmdvl posted provides a lot of good information on miniature cats. (There actually are some and the process started about 170 years ago.) The article notes the difficulty in obtaining the “correct” size, a difficulty that dog breeders were addressing hundreds of years earlier.
Beyond that, note the size of the miniature dogs: about the size of cats. If the impetus for the small dog breeds was to have a “lap” dog, then cats started out with the advantage that they were already the right size, so there was less lure to do anything to change them. (Human fancy being the odd feature that it is, eventually someone did decide to breed miniature cats, just as I am sure that someone will soon begin attempting to breed Rodents Of Unusual Size, but you need sufficient numbers of odd people to support such activities which required the larger world populations (and larger middle and wealthy classes) to find sufficient numbers of such people.)
Mini-cats already exist; they’re called “ferrets”.
Maybe ask around to see if any litter has a “runt?” Maggie’s still not much bigger than she was as a kitten.
Oh, and I’m NOT debating you, but if you keep a cat indoors from the time he or she is a kitten, Kitty shouldn’t be at all unhappy.
Miniature cats are just far too puntable.
Or, it might be a little fairer to say that the historical function of cats as exterminators was something that was already part of their instinctive behavior, so there was little incentive to breed different sizes. Given how formidable the largest rats can be, there would have been no incentive to breed miniature cats.
You’ve got t help me out here, because I drink my beer so quickly, it never has time to get fully grown. How big does it grow? 
Tangetially related… I visited my college roomie’s home in rural Oklahoma many years ago. They had a cat who had kittens and a neighbor who didn’t like cats. So the neighbor put a pan of engine coolant underneath his porch and a lot of the kittens got into it and died. They managed to save one kitten who was in the pan at the time lapping up the coolant. The kitten survived but its growth was stunted and looked like a kitten as an adult.
Do you think we could easily breed domestic cats to come in a variety of shapes and sizes like dogs? Is there anything we know about the feline genome that would make it less amenable to selective breeding than the dog genome? I think the OP raises an interesting question, especially since most dog breeds are only a few hundred years old…
There are, sadly, some rather bizzare intentionally bred variations out there. I don’t know why anyone would want them, but they’re out there. The Munchkin cat has very short legs. The Twisty cat is normal except for extremely short, useless front legs. The Ragdoll cat has very little muscle tone, and can be draped around like a limp pillow. All of these have a low survival rate in the womb and in kittenhood.
Then there’s the Peke-faced Persian. It has been bred to win cat shows, and it has a flat, broad face with an almost non-existant snout. It cannot eat normally; it has to lap up pureed food. It has no passages for the tears to flow into the nose, so the owner has to continually wipe the face. Many of this kind of cat must be on antibiotics from womb to tomb.
In my opinion, it is animal cruelty to perpetuate these varieties.
Dogs are much more variable genetically than other domesitic animals. This paper (caution: pdf) attributes this in part from the antiquity of domestication, plus the fact that dog genes were drawn from at least 4 different wolf populations. (It is uncertain whether or not these represent independent domestication events, or just instances of admixture with wolf populations in different areas). Even most breeds of dogs have an unexpected level of variability.
I am not aware of any intrinsic genetic characteristics of the species ancestral to domestic cats that would make it harder to selectively breed. The differences may be because (1) cats were domesticated much more recently than dogs; and (2) they originated from fewer domestication events, and/or had fewer if any instances of backcrosses to wild populations (or the backcrosses were in a more restricted area). Besides that, as has already been said, (1) there has been less motivation to selectively breed cats, (2) selective breeding began more recently, and (3) cats by their nature are more difficult to selectively breed since mating often takes place out of human control.
Of course, cheetahs are known for having extremely low genetic variability, but as far as I know this not the case with other cat species.
Interesting. I thought the most commonly accepted theory was that all dogs were descended from one population of wolves in (IIRC) Central or South Asia. But I guess we’ll go back and forth between a few competing theories before one finally wins out.
This is because the population of cheetahs is extremely small, causing considerable inbreeding. That’s unlikely to be true for domestic cats.
I was surprised by that too.
It’s not so much because the present population is small - it was much larger, and the species had a much larger range, during historical times - but that the species is thought to have undergone an extreme population bottleneck in the past, probably in the late Pleistocene. Cheetahs are all so similar genetically that they can accept skin grafts from one another. Inbreeding in modern populations may cause addition problems, but it is not at the root of the situation.
Domestic cats, and as far as I know all other species of cats, are much more genetically variable than that.
I think that’s the wrong link Looks like the dog genetic diversity link.
Yes, I’ve heard it said that all extant African cheetahs are essentially identical twins. That might be a bit of an exageration, but it aligns with the skin graft idea.
Sorry, here’s the link I meant to post.
Cheetahs are not completely genetically identical, so they are not as close as identical twins, but they do have extremely low genetic variability.
Even though the modern population of domestic cats is very large, it would be possible for it to have low genetic variabilty if the founder population had been very small and remained that way for a number of generations (and didn’t back-cross to the ancestral population). This does not seem to have been the case.
While this doesn’t mean terriers weren’t specially bred, I would like to point out that Farley Mowatt reported in Never Cry Wolf that the wolves he observed lived off of mice for much of the year. So in one sense, the ancestors of dogs were killing mice comparably to cats. I think we have to look elsewhere for the root cause of the difference between the proliferation of dog breeds and the relative sameness of cat breeds.
Sailboat
However, wolves stalking field mice on the prairie (or even the forest) is a bit different than keeping mice and rats out of a granary. Even a diminutive wolf would be hard pressed to get into the nooks and crannies (and interior walls) of a building constructed to hold grain, while cats can “patrol” such places quite easily.
The runt won’t necessarily stay all that small - my old roommate had the runt who turned out to be 10-11 pounds full grown, not overweight. Not that huge for a cat, but he certainly didn’t stay runty.
My OP was probably more conjecture than anything. I’m not actively looking for a cat, small or large. In fact, at the risk of offending some people, I’m one of those folks who absolutely loathes small dogs - horrible, shivery, needy, yappy, rodent-like things, so it may be that I’d see the sort of cat I described and have the same reaction.
Off topic, I was only able to think of six (lion, tiger, leopard, cheetah, jaguar, puma) unless you’re counting Siberian tigers and snow leopards as seperate beasts. Which they may well be, or I may be “drawing the line” too high and excluding things like ocelots, I was just curious to hear your eight.