The UK is big on Shetland Pony racing. This example is a miniature steeplechase format, with child riders in jockey's silks, and shaggy little ponies (Shetlands are typically 10-12 hands high!) flying along over the hurdles with ease. Very cute.
The South African Boer Horse (Boerperd) descends from an interesting mixture of breeds. The first horses brought to South Africa by Europeans were Andalusians, Persian Arabs and Java Ponies. These were mixed and selected to provide all purpose horses and cavalry mounts. In the 1800s the blood of Friesians, Norfolk Trotters, Cleveland Bays and Hackneys was added. Today's Boerperd is somewhat variable in appearance, with some horses showing their Arabian or Andalusian ancestry more strongly, and others leaning more towards the Norfolk Trotter, Hackney or Cleveland Bay appearance.
As can be seen on this website (look under "Stallions/Hingste") some of the horses are 5-gaited, and shown saddleseat style. Others are clearly trotting horses with sport-horse conformation, and are used as jumpers or for dressage. Others are used as pleasure horses.
Because of the many influences, the Boerperd comes in a variety of solid colors, including roan and dun.
These British mounted police do regular performances showing the incredible discipline and steadiness of their fabulous horses. Whether it's jumping through rings of fire or removing the entire saddle while riding a steady canter (wait for it at the end of the video!) these horses and riders are really impressive. Horses selected for police work have to deal with crowds, traffic, noise, gunfire and all sorts of scary things. Only the most steady and brave graduate from the training programs.
I saw a Mexico City mounted police display once, and they did some nifty acrobatics, such as four riders galloping in a group, swooping down on a fifth guy, with each rider grabbing either an arm or leg and carrying the man around like an airplane. I'll dig up some more police horse videos in the future. The level of trust these horses display is really something.
Update: Here is the official website of the Metropolitan Mounted Police (of London, UK), which has interesting info about how they select and train the horses.
Puissance is a show jumping competition where horse and rider attempt to clear a very high solid wall, often over 7 feet high. The solidity of the wall makes it psychologically intimidating, as the horse cannot see what lies on the other side. Like all jumps in show jumping, the wall is not really solid - it is made of stacked blocks that will fall away at a light touch. This means a light touch might knock the top off, making you lose, but it also helps prevent injury should the horse and rider misjudge the jump. Nonetheless, a bad landing can cause serious injury to horse or rider.
All horses and ponies can jump a few feet (an important skill when escaping predators in the wild), but only a few are physically and mentally capable of being trained to jump at a competitive level.