Java Edition:Super Horse
In 13w16a (1.6 snapshot) when horses were first introduced their maximum speed was 2.0 (84.32 blocks/second) and their maximum jump height was 2.0 (16.375 blocks). Also the minimum jump height was 0, so you can get a horse that is unable to jump.
However, due to a bug from 13w16a to 13w18b, horses would despawn similar to how hostile mobs despawn. Due to horses only spawning via the generation of new chunks, horses would essentially despawn immediately due to its distance from the player. This means to obtain a horse the player would either have to spawn into a new world within a plains biome as that would cause horses to spawn in close proximity, or alternatively create a nether portal in the nether that would take the player to a plains biome.
Furthermore, a bug in the server version of 13w16a would cause horses to be untameable.[1] This is because a player riding any ridable entity such as a boat, minecart or horse would immediately kick the player from the server and prevent the player joining back. This means that the player would have to upgrade to 13w18c or later to be able to keep and tame the horse.
These horses will lose their speed statistics upon updating into 13w21a and beyond, as the movement to the new attribute system did not include a data fixer to transfer the data for speed. Any existing horses will be assigned a random speed. Health and jump height will be preserved.