The American Shorthair cat is not rare in an absolute sense. It's a regular purebred cat born out of a standard American moggy. This cat breed is a refined American moggy which is great and it is a very good purebred cat suited to indoor living and good with kids. But it's impossible to consider the cat as being rare.
|American Shorthair - classic tabby. Photo copyright Helmi Flick.|
That said, the meaning of the word "rare" is elastic. In terms of the overall number of domestic cats in America you might argue that the American Shorthair is quite rare. However, in an absolute sense it is not rare. You will see them in households around America.
You will not see cats of this breed probably in Europe that much whereas you will see the more popular and perhaps better known American cat breed, the Maine Coon, more frequently in Europe. So perhaps you might argue that the American Shorthair cat is relatively rare in Europe but much less so in America, the home of their origin.
As it happens, I have a very comprehensive page on how rare the cat breeds are. I wrote it many years ago but I made a big effort to be accurate and used various criteria to decide on the rarity of the breeds. If you click on this link you'll be taken to the conclusions of that page where there is a list of what I consider to be the rare cat breeds.
Apparently the number one cat association in Britain, the GCCF, believe that the Sokoke is the rarest cat breed. It is within my list as well but there are others. Some of the breeds I've listed have been discontinued and therefore there may be no cats of these breeds left in the world. That happens sometimes: cat breeds die out and therefore the cats of the breed disappear eventually.