It's hard to know how many specific dog breeds exist -- the World Canine Organization recognizes nearly 350; the American Kennel Club recognizes fewer than 200. A vast number of them are short-haired, but it's not obvious: Some have an undercoat, and some do not. Some have thick pelts and some thin. Some have smooth coats and some short-hair coats are fuzzy.
Short Coat, Double Coat
Short-hair dogs with significant undercoats come in all sizes, from German shepherds and Labrador retrievers to Welsh corgis and border terriers. Like all double-coated dogs, they will "blow coat," or shed massively, one, two or even three times a year, requiring regular grooming for them and extra housecleaning for you. Single-coat dogs, by the way, still shed, but they do so continuously and in less volume.
Thick Coat, Thin Coat
A short coat can be thick or thin. The Shiba Inu and the Belgian Malinois have some undercoat but are less fluffy than the true Arctic types such as the Pomeranian and the Alaskan malamute. The Australian cattle dog has a short coat that can feel thick and slightly rough. The saluki has short, soft hair all over except for long, silky hair on the long ears, the legs and the tail. Some breeds have so little hair as to be considered hairless. These include the partially haired Chinese crested -- but not the "powder puff" type -- the American hairless terrier, and the truly hair-free Peruvian Inca Orchid and the Xoloitzcuintli, also known as the Mexican hairless.
Smooth Coat, Single Coat
A smooth single coat is common in many breeds, from the gigantic Great Dane and the various mastiffs to the tiny Chihuahua, and it occurs in all the AKC groups from hounds to sporting dogs such as the pointers and some herders including the smooth collie. Medium-size smooth-coats include the bulldogs, the boxer, the rottweiler, pit bulls and similar Staffordshire and American Staffordshire terriers, Doberman pinschers, and greyhounds and whippets. Smooth-coat toys are numerous: the smooth fox terrier, the rat terrier, the miniature pinscher, the Boston terrier, the pug and the miniature dachshund. These dogs require little grooming other than a going-over with a soft brush about once a week or a swipe with a cloth. However, they still shed, just as you do.
Short Coat, Not Smooth
Just because a dog's coat is short doesn't mean it has to be smooth. Different hair textures occur in many breeds, as well as within breeds. The German shorthaired pointer comes in both a smooth and a rough version. The Chesapeake Bay retriever's coat is both rough and wavy, as well as characteristically oily. A Jack Russell terrier with a broken coat looks like he has a serious case of the frizzies, with long, curly hairs spiraling in all directions. The rough variety of the Brussels griffon can look like he's just having a bad hair day all round, and this coat has been described as "ungroomable."