Here's a long list. I'll add some images when I've found them all.
Abaco Barb Horse
Abtenauer
Abyssinian Horse
Aegidienberger
Akhal-Teke
Albanian Horse
Altai Horse
Alter Real
American Cream Draft
American Crème and White
American Paint Horse
American Quarter Horse
American Saddlebred
American Spotted Paso
American Walking Pony
American Warmblood
Andalusian
Andravida Horse
Anglo-Arabian
Anglo-Kabarda
Appaloosa
Arabian
Arappaloosa
Ardennes Horse
Arenberg-Nordkirchen
Argentine Criollo Horse
Ariegeois Pony
Asturcon
Asturian
Australian Brumby
Australian Pony
Australian Stock Horse
Auxois
Avelignese Pony
Azores Horse
Azteca Horse
Baise Horse
Balearic Horse
Bali Pony
Balikun Horse
Baluchi Horse
Ban-ei
Banker Horse
Barb Horse
Bardigiano Pony
Bashkir Curly
Bashkir Pony
Basotho/ Basuto Pony
Basque Pony
Batak Pony
Bavarian Warmblood
Belgian Horse
Belgian Warmblood
Bhirum Pony
Bhotia/ Bhutia Pony
Black Forest Horse
Boer Pony
Bosnian Pony
Boulonnais
Brabant Horse
Brandenburger
Brazilian Sport Horse/ Brasileiro de Hipismo
Breton Horse
British Riding Pony
Buckskin
Budyonny/ Budenny Horse
Buohai Horse
Burmese Pony
Buryat Horse
Byelorussian Harness
Calabrese Horse
Camargue Horse
Campolina
Canadian Cutting Horse
Canadian Horse
Cape Horse
Carolina Marsh Tacky
Carthusian Horse
Caspian Pony
Cayuse
Cheju Pony
Chilean Corralero
Chilean Horse
Chilote Horse
Chincoteague Pony
Chinese Guoxia
Cleveland Bay
Clydesdale Horse
Colonial Spanish
Colorado Ranger Horse
Comtois Horse
Connemara Pony
Criollo Horse
Crioulo
Dales Pony
Danish Warmblood
Danube
Dartmoor Pony
Deliboz
Djerma
Dole Gudbrandsdal
Dole Trotter
Don
Dongola
Dülmen Pony
Dutch Draft
Dutch Heavy Draft
Dutch Warmblood
East Bulgarian
East Friesian Horse
Egyptian
Eriskay pony
Estonian Native
Exmoor Pony
Faeroes Pony
Falabella Horse
Fell Pony
Finn Horse
Fjord Horse
Fleuve
Florida Cracker Horse
Fouta/ Foutanké
Frederiksborg
Freiberger
French Saddle Pony
French Saddlebred
French Trotter
Fresian Horse
Galiceño
Galician Pony
Garrano
Gelderlander Horse
German Riding Pony
Gidran
Golden American Saddlebred
Gotland Pony
Groningen
Guangxi
Gypsy Vanner Horse
Hackney Horse
Hackney Pony
Haflinger Horse
Hanoverian Horse
Heck Horse
Hequ
Highland Pony
Hispano Horse
Hokkaido Pony
Holsteiner
Hucul Pony
Hungarian Warmblood
Icelandic Horse
Indian Half-Bred
Iomud
Irish Draft
Irish Sport Horse
Italian Heavy Draft
Java Pony
Jinzhou
Jutland Horse
Kabarda Horse
Karabair
Karabakh Horse
Kathiawari
Kazakh Pony
Kentucky Mountain Saddle Horse
Kerry Bog Pony
Kiger Mustang
Kinsky-Horse
Kirdi Pony
Kisber Felver
Kiso
Kladruby/ Kladruber
Knabstrup
Konik
Kushum
Kustanair
Landais Pony
Latvian Harness Horse
Latvian Snow Pony
Lipizzaner
Lithuanian Heavy Draft
Lokai
Losino
Lundy pony
Lusitano
Lyngshest
M’Bayar
Malapolski
Manchurian Pony
Mangalarga
Mangalarga Marchador
Manipuri Pony
Maremmana
Marwari Horse
Mérens Pony
Messara
Miniature Horse
Misaki
Missouri Fox Trotting Horse
Miyako
Mongolian
Morab
Morgan Horse
Moyle
Murakoz Horse
Murgese
Mustang Horse
National Show Horse
New Forest Pony
New Kirgiz
Newfoundland Pony
Nez Perce Horse
Nigerian Pony
Nokota Horse
Noma Pony
Nonius Horse
Nooitgedacht Pony
Nordland
Nordlandshest
Noric
Noriker Horse/ Pinzgauer
North Swedish Horse
Northeastern
Northlands Pony
Norwegian Fjord Horse
Novokirghiz
Ob Pony
Oldenburg Horse
Orlov Trotter
Paint
Palomino
Pantaneiro
Paso Fino
Peneia
Percheron
Peruvian Paso
Pindos Pony
Pinia
Pintabian
Pinto
Pleven Horse
Poitevin Horse/ Mulassier
Polish Konik
Poney Mousseye
Pony of the Americas
Pottok
Przewalski’s Horse
Pyrenean Tarpan
Qatgani
Quarab
Quarter Horse
Quarter Pony
Racking Horse
Rhenish-German Cold-Blood
Rocky Mountain Horse
Russian Don
Russian Draft
Russian Heavy Draft
Russian Trotter
Sable Island Pony
Salerno
San Fratello Horse
Sandalwood Pony
Sanhe
Sardinian Anglo-Arab
Sardinian Pony
Schleswiger Heavy Draft
Schwartzwälder Fuchs
Selle Francais
Shagya
Shagya Arabian
Shales Horse
Shetland Pony
Shire Horse
Single-Footing Horse
Skogsruss
Skyros Pony
Sokolsky Horse
Somali Pony
Sorraia
Soviet Heavy Draft
Spanish Mustang
Spanish Tarpan
Spanish-Barb
Spanish-Norman
Spiti Pony
Spotted Saddle Horse
Standardbred
Sudan Country-Bred
Suffolk Punch
Sumba and Sumbawa Pony
Swedish Ardennes
Swedish Warmblood
Swiss Warmblood
Syrian Horse
Taishuh
Tarpan
Tawleed
Tennessee Walking Horse
Tersk
Thessalian
Thoroughbred
Tibetan Pony
Tiger Horse
Timor Pony
Tokara
Tori Horse
Trait Breton Horse
Trait Du Nord
Trakehner
Ukranian Riding Horse
Ukranian Saddle
Virginian Highlander
Vlaamperd
Vladimir Heavy Draft
Vyatka
Waler Horse
Walkaloosa
Welara Pony
Welsh Cob
Welsh Pony
West African Barb
Western Sudan Pony
Westphalian Horse
Wielkopolski
Württemberger
Xilingol
Yakut
Yanqi
Yili
Yonaguni
Zaniskari Pony
Zemaituka/ Zhumd
Zhemaichu