GENETIC BASES FOR COAT TYPES
Most of the genetics behind coat types aren't known or are too complicated to go into here. But I will just say a little bit about hairless dogs and longhaired dogs, in case you're interested!
HAIRLESSNESS
There are four hairless breeds of dog - the Mexican Hairless (Xoloitzcuintle), Peruvian Inca Orchid, American Hairless Terrier and Chinese Crested Dog.
The Xolo, Inca Orchid and Chinese Crested all have dominant hairlessness, so they only need one copy of the gene to be hairless.
In fact, all hairless dogs of these breeds are heterozygous for hairlessness, because it is one of the only known examples in dogs of a lethal gene. This means
that a homozygous dog (one with two copies of the hairless gene) will miscarry.
Because all hairless dogs of these breeds are heterozygous, breeding a hairless dog to a hairless dog will result in some puppies which miscarry (the doubles), some which
are normal hairless dogs (the singles), and some which are not hairless at all. Remember our Punnet squares for the merle gene? Mm x Mm results in one
MM (double merle), two Mm (normal merles), and one mm (non-merle). This is the same for hairless dogs.
The coated dogs which appear in hairless litters are often not allowed to be shown. The exception is the coated Chinese Crested, which is known
as the "powderpuff" variety, and is accepted by kennel clubs.
The American Hairless Terrier is very unusual indeed, because it shows the only known example of recessive hairlessness in dogs. Recessive hairlessness does just what it says on the tin - a dog must have two copies of the gene in order to express it. It is not a lethal gene, unlike dominant hairlessness, and because all hairless AHTs are homozygous, coated puppies will never appear when a hairless dog is bred to another hairless dog.
Both types of hairlessness come with associated health problems. These are mainly skin conditions and allergies, and obviously, dogs with no hair are highly likely to get sunburn and skin cancer unless they are protected from the sun. Dominant hairlessness can also cause tooth problems (usually less molars than normal).
LONG HAIR
The gene, or at least one of the genes, causing long hair in dogs is recessive. The main points of interest here are that it means a longhaired mixed breed must have two parents who are longhaired or long hair carriers (e.g. a Labrador x Golden Retriever would be shorthaired, so a longhaired Retriever mix must have Golden on both sides or some other longhaired breed), and the gene can also be carried down many generations without being expressed, which means that many shorthaired breeds occasionally produce longhaired puppies.
This is an example of a longhaired Shar Pei, known as a "bear coat". Such dogs turn up in shorthaired litters relatively regularly.