These are sourced to the best of my ability from real breed registries and The Equine Tapestry (book and website). There may be some mistakes as registries do not always use consistent terminology (most European drafts will register "palominos" when no cream gene is present, they are referring to a light flaxen chestnut; etc). I will try to post where different registries allow different colors or not all colors that exist in the breed. Please comment with corrections (with links to sources, please) and suggestions for which breeds to add next.

Terminology:

Allowed = A horse with this color may be registered with at least one breed registry.
Disallowed = A horse of this color is not allowed to register with at least one breed registry.
Documented = There is genetic test results or unambiguous photographic evidence of the color occurring in a pedigreed, purebred horse.

Colors that may appear in any breed:
Manchado: Appears to be environmental, has occurred in several breeds (TBs, Arabians, Criollo, Hackney) but only in horses bred in Argentina.
Chimera: The result of two embryos fusing, causing patches of "foal 1" and "foal 2"'s coat colors where each's respective DNA is expressed.
Brindle: Appears to be a form of chimerism and not thought to be hereditary.
Somatic mutation: A gene may be accidentally "turned off" in a portion of the skin resulting in odd colorations. Not hereditary.


Akhal-Teke: Black, chestnut, bay, grey, cream, dun. All white patterns are allowed, but only sabino has been documented, though blue eyes do occur. This breed has a characteristic "shine", but it is not to be confused with the champagne color, which is not present in this breed.
American Cream Draft: Chestnut base with Champagne dilution. Cream is extremely rare but does occur in the breed, cremello horses have not been documented.
----Notes: Champagne is not homozygous in the breed, so purebred chestnut foals do occur and may be registered as tracking ACDs.
Andalusian: The international registry accepts bay, black, chestnut, grey, cream, and dun but will also accept other colors with confirmed pedigree. Roan, sabino, silver, and pearl are rare but have been documented.
Appaloosa: Chestnut, black, bay, grey, roan, rarely silver or champagne, sabino allowed, overo and splash documented but not allowed, varnish/blanket/leopard required.
Arabian: US Registry accepts black, bay, chestnut, grey, "roan" (probably rabicano). Documented in black, bay, chestnut, grey, rabicano, sabino, dominant white.
Ardennes (French): Bay, chestnut or black (rare and not preferred) base color, often come in roan, flaxen or mealy, occasionally grey.
Ardennes (Swedish): Allowable in any color, documented in bay, chestnut, black, roan, grey, silver ???
Australian Stock Horse: Bay, chestnut, or black base. Grey, roan, cream and silver possible. Tobiano, sabino, and overo possible.
Auxois: Bay, chestnut or black (rare) base color, often come in roan.
Bashkir Curly: Chestnut, black, bay, grey, roan, cream, dun, silver, champagne, toabiano, sabino, rarely overo or splash.
Belgian (American): Chestnut, sometimes mealy or flaxen, rarely roan. Recent Brabant imports have reintroduced bay but only a handful exist in the US.
Black Forest Horse: Breed registry says only liver chestnuts, but it is theoretically possible that the breed is a uniform Silver Bay. To my knowledge genetic testing has not been done.
Boulonnais: Chestnut base, rarely black, almost exclusively grey (One non-grey stud exists)
Brabant: Black, Bay or Chestnut, often with roan. Silver is suspected but not documented. Grey is possible.
Breton: Commonly chestnut. Also bay, black, roan, or grey.
Caspian: Chestnut or bay, rarely black. Also grey and sabino.
Cleveland Bay: Bay only.
Clydesdale: Most commonly bay, also black or chestnut. Grey is allowable in some breed registries. Sabino is present in all animals, "sabino roan" is not uncommon, "true roan" does not exist.
Comtois: Most commonly flaxen chestnut, also bay.
Connemara: Typically black or bay, rarely chestnut. Grey, roan, cream, dun, silver, and sabino possible.
Criollo: Chestnut, bay, black, grey, roan, cream, dun, sabino. Varnish appaloosa possible, but rare.
Dartmoor: Typically black or bay, chestnut possible. Can be grey or (rarely) roan.
Dutch Heavy Draft: Most chestnut or bay, rarely black. Most are roan, grey is allowable.
Exmoor: Bay only.
Fell: Most commonly black, also bay or grey. White markings are discouraged.
Finnish Draft:: Most commonly flaxen chestnut, rarely bay or black. Cream, Roan, Rabicano, Sabino, and Silver are recorded but rare.
Frederiksborg: Chestnut, bay, rarely black. Cream and sabino possible.
Friesian: Only Black. Chestnut exists, but either may not be registered or may not be registered as a breeding animal. Grey exists in animals which were crossed to a grey Arabian and crossed back over several generations, but these animals are not purebred or able to be registered as Friesian.
Gelderlander: Chestnut, bay or black base. Grey, cream and silver possible.
Georgian Grande: All colors allowable if present in foundation breeds (American Saddlebred and draft breeds)
Gotland: Bay, chestnut, rarely black base. Cream and splash possible.
Groninger:  Bay or black, rarely chestnut. Grey, tobiano and sabino are possible, but rare.
Gypsy Drum: All colors allowable, most typically black tobiano. Chestnut, bay, black, tobiano, overo, sabino, splash, roan, cream, dun, pearl, varnish, leopard and blanket are possible via crossbreeding. Champagne not documented.
Gypsy Vanner: All colors allowable, most typically black tobiano. Chestnut, bay, black, tobiano, overo, sabino, splash, roan, cream, dun, pearl, varnish, leopard and blanket have been documented. Champagne not documented. (The stallion Taskin is NOT a champagne, he is a buckskin)
Hackney: Bay, black or chestnut base. Roan and sabino possible.
Haflinger: Always chestnut, can be liver, mealy, and/or flaxen. NO palominos.
Hanoverian: Black, bay or chestnut base. Grey is common. Sabino is documented, but prohibited in the registry.
Highland Pony: Rarely chestnut, usually bay or black. Grey, dun, and silver possible.
Icelandic: Bay, black or chestnut base. Grey, roan, cream, dun, silver, tobiano, splash, and dominant white possible.
Irish Draught: Any solid color allowable by the registry, most commonly grey. Documented colors: chestnut, black, bay, grey, dun, sabino. No roan.
Italian Heavy Draught: Chestnut or bay, commonly roan.
Jutland: Typically chestnut, may be bay, black, or roan.
Kathiawari: Bay, black, or chestnut base. Grey and dun are rare but present. Tobiano, sabino and splash are documented.
Kladruber: Black base only. Sabino and grey are present.
Knabstupper: Black, chestnut or bay base. Grey present but not preferred. Cream is rare, sabino is present. Varnish is extremely common, leopard is more common than blanket.
Lippizanner: Bay or black base, rarely chestnut. Usually grey.
Lusitano: The international registry accepts bay, black, chestnut, grey, cream, and dun but will also accept other colors with confirmed pedigree. Roan, sabino, and pearl are rare but have been documented.
Marwari: Bay, black, or chestnut base. Grey and dun are rare but present. Tobiano, sabino and splash are documented.
Noriker: Bay, black, or chestnut base. Roan, tobiano, and Leopard patterns are observed. Overo is listed as possible but no photographic proof has been found.
Norman Cob: Most commonly bay or chestnut, grey exists.
North American Spotted Draft: Any solid color allowed with Tovero or Overo. No appaloosa markings. Studbook is open to Gypsy Drums and American Cream Drafts, so all genes are possible.
North Swedish Draft: Bay, brown/seal bay, black, flaxen chestnut, cream, dun, and grey in certain pedigree lines. Double cream dilutes may not be used for breeding.
Norwegian Fjord: Breed is exclusively dun, though not entirely homozygous (non-duns are threoretically possible but haven't been seen for many generations) Base colors are 90% bay. Chestnut, black, and cream are present but rare.
Miniature Horse (American): Black, bay or chestnut base. Grey, roan, cream, dun, silver, champagne, tobiano, sabino, overo, splash, and appaloosa genes present.
Morgan: Bay, chestnut or black base. Grey, roan, cream, dun and silver possible, but rare.
Paint: Black, bay or chestnut base. Grey, roan, cream, dun possible. Champagne and pearl possible but rare. Must have tobiano, overo, sabino or splash.
Paso Fino: Bay, black, or chestnut base. Grey, roan, cream and dun possible. Tobiano and sabino are possible.
Percheron: Mostly grey or black. In the US chestnut, bay and roan are allowable.
Poitevin: Grey, bay, black, or dun.
Quarter Horse: Bay, black, or chestnut base. Grey, roan, cream, dun, possible. Silver, pearl and champagne possible but rare. Tobiano, overo, sabino, dominant white and splash are possible but only allowed in some registries. Apaloosa/Varnish markings are possible. Not all of these colors are allowable in many registries.
Saddlebred (American): Any color may be registered, but only chestnut, bay, black, grey, cream, silver, champagne, tobiano, sabino, overo, and splash have been documented.
Selle Francais: Bay is more common than chestnut or black. Grey and sabino are possible, tobiano is rare.
Shire: Black, bay, or grey allowed universally. Roan allowed in geldings and mares only. Chestnut allowed in the US only. NO silver; white mane and tail may be caused by the sabino gene. Sabino is definitive and found in all horses. Excessive white is discouraged. Cream exists in the US due to a time when the studbook was temporarily opened, however it is only documented in one or two horses and many do not consider them to be purebred though they are registered.
Suffolk Punch: Always chestnut, commonly flaxen.
Sugarbush Draft: Any solid color, with appaloosa markings preferred but not required as long as the horse has appaloosa heritage. No paint markings.
Thoroughbred: Chestnut, bay or sometimes black. Grey is common. Cream is rare. Sabino, overo, splash and dominant white have been documented.  
Trakehner: Chestnut, bay, or black base. Grey and roan are possible, as are tobiano and sabino.
Vladimir Heavy Draft: Most commonly bay, also black or chestnut.
Welsh Cob: Black, Bay, Chestnut, dun, cream, grey, roan, sabino and splash are documented.