What is the name of the smallest breed of horse?

The Falabella is the horse breed commonly recognized as the smallest breed of horse.

The Falabella is an Argentine miniature horse developed from local Criollo stock beginning in the 19th century. Patrick Newtall started the breeding program around 1868, and his son-in-law Juan Falabella later continued it. Selective breeding, including the use of Shetland ponies, Welsh ponies, and small Thoroughbreds, produced a consistently tiny horse with the proportions and refinement of a full-sized horse rather than simply a pony-shaped animal.

Adult Falabellas commonly stand roughly 28 to 34 inches at the withers, though exact standards vary among registries. They are generally kept as companion animals, exhibition animals, or trained performers; their size makes them unsuitable for ordinary adult riding.

“Miniature Horse” is a broader type or registry category, not the single breed name the question seeks. Shetland and Welsh ponies are also small, but they are substantially larger than a Falabella.