Ponies will often identify with a specific breed of (bio-)horse that they feel embodies their personality. Some ponies will identify with other real-life equids (e.g. zebras, donkeys), and other ponies will identify with fictional and mythical equids (e.g. unicorn). I have compiled a brief list covering what I believe to be the most common breeds (both fictional and real) encontered in ponyplay.
American Quarter Horse - As their name implies, quarter horses are well suited to short distance sprinting (quarter mile) and, like Thoroughbreds, are used as both race horses and general sport horses. While probably most known for their dominant role in many western disciplines, they can also be found in many english disciplines as well.
Arabian - An old breed known for its endurance, unique head shape, and high tail carriage, Arabian bio-horses have a rich history and hold a strong influence on the bloodlines of other breeds such as Thoroughbreds and the American Quarter Horse.
Clydesdale - A breed of draft horse with the Budweiser Clydesdales probably being the most famously known examples of the breed.
Donkey - Donkeys are not a breed of horse, but they are equids.
Thoroughbred - A breed of horse most commonly known from horse racing, Thoroughbreds are hot-blooded and associated with speed and agility. Outside of racing, Thoroughbreds are trained and compete in many english disciplines.
Unicorn - A mythical creature that has the overall form of an equid but with a distinctive spiralling horn emanting directly from the forehead.
Zebras - They are not a breed of horse, but they are equids, and there are a few human ponies that identify with zebras (either primarily or as an alternate pony persona). Zebras are fairly small: they are typically 12.2-13 hands, which is well under the commonly used definition of a pony (mature horse that is 14.2 hands). Their distinctive coloring and wild nature are two characteristics that separate them from domesticated horses.