Goldfish have been bred for centuries. Because of their great variety, they can be difficult to classify. The easiest way is through fin arrangement. Click on the picture for each breed to see an expanded entry for it.
All existing breeds can be classified into one of three groups based on their fin arrangement:
Single Tailed Goldfish
These are goldfish that still look similar to the normal carp they are decended from except for their color (and maybe the shape of their fins). Single-tailed goldfish do not normally exhibit traits such as portruding eyes or head growth, though some sport varieties might have these traits.
All single-tailed varieties have a regular carp-shaped body.
Double Tailed Goldfish
These are normal goldfish that have a split tail. The fish basically has two tails that split off at the base of it's spine. Occasionally the anal fin (which is normally a single fin) will be split as well.
Some Fantails will have a conventional carp-like body, but most are egg-shaped, torpedo shaped, or teardrop shaped. But the thing they have in common is that all of them have split tails.
Dorsalless Goldfish
These are Fantailed fish that lack a dorsal fin. All Dorsal-less goldfish have egg-type bodies and lack a dorsal fin. They are expected to have no bumps or spikes in the back arch (bumps are basically vestigial dorsal fins).
Catagory Criteria
These are the criteria for each entry. When available, I have included links to YouTube videos for some breeds as well. The following items are listed under each entry:
Difficulty
How difficult the fish is to keep.
Beginner fish are extremely resiliant and can survive a lot of punishment (good for kids). They can be kept in Aquariums or bowls and can survive a broad range of temperatures. Beginner fish are capable of living in outdoor ponds once they reach adulthood.
Intermediate fish are more delicate, and require extra attention. They should be kept only in Aquariums with a temperature of 55 degrees or more.
Advanced fish are very fragile, and will die easily if you dont know what you're doing. They should be kept only in Aquariums which are maintained carefully, and should only be kept with other breeds that are similarly delecate, to ensure that they can get their share of food.
These are general guidelines intended to inform beginners. Any breed of goldfish could be kept in a pond (for example) if the person keeping them knew what they were doing.
Availability
This indicates how likely your local pet store is to have the fish in stock.
Common fish are available at all pet stores that sell fish. You can probably find these in many non-pet stores (such as walmart) as well.
Uncommon fish will be available at only at chain pet stores (like Petsmart or Petco), and shops that specialize in fish.
Rare fish may be occasionally be available in storefronts that specialize in fish, but will most likely only be available over the internet or by mail order.
Very Rare fish may not be available at all, or only from certain breeders (and will be very expensive). These are almost always mail order.
Price
This is the typical price of a small fish at a common pet store in the US (or on the internet). By "small" I mean between 1 and 2 inches in body length (not including fins).
Very Cheap fish include anything under $1 (usually fish intended as feeders for other fish).
Cheap fish include anything under $5...unlike Very Cheap fish, these are intended as pets and not as food for other fish.Moderate fish are anywhere from $5 to $15.
Expensive fish are anything over $15...some of these are simply adult versions of cheap fish. Because of their size, they are worth more money.
Very Expensive fish are show quality specinmens that can run into the hundreds of dollars. These are the fish that are typically bought as breeding stock.
The prices for most small fancy goldfish are under $10 each. Most non-show quality specimens will max out at around $50 for large adults.
Show quality fish (which includes many of the fish pictures on this site) can usually be bought only from foreign breeders over the internet. Expect to pay $30 to $60 in shipping (which has to be overnight, since they are live fish), in addition to $30 to $200 or more per fish. Small fish bought in this manner are typically only available in bulk.
Fish bought over the internet in this way are usually insured in case they die in transit. But be sure to read the breeder's shipping and return policy before you buy it.
Adult Size
An estimate on how large (in inches) the fish is expected to get. The more exotic breeds tend to be smaller. The estimate is approximate, and is conservative.
Scales
Most goldfish scales are metallic, but some also come in nacreous ("mother of pearl") and matte (nonreflective) varieties.
The scales are transparent regardless; the color comes from the skin under the scales, not the scales themselves. Reflective scales appear more metallic. Non-reflective scales appear matte.
Special Care
Overview of the additional care the breed requires.
History
Ideal body shape
This is a chart showing the different areas of the body on a goldfish:

The four main body types are:
Wild type bodies are the default carp-like shape, usually long and slender.
Torpedo shaped bodies are more cylindrical.
Egg type bodies are wide and deep.
Teardrop bodies are so deep as to be almost spherical.
Body "depth" is the measurement of the vertical dimension of the fish, and is usually relative to the length (none of these breeds should have a depth larger than the length). Body dimensions never include fins. The details of the hood or other body parts will also be in this section.
Hoods
If the fish has a hood, the ideal size and shape will be listed in the "body" section of the text.
The hood (or "wen") is the spongy growth on the heads of some breeds. This growth is present on all goldfish to one degree or another, but has been bred to be much more prominant in some breeds. Partial hoods cover the top of the head (looking like an exposed brain). Full hoods cover the entire face and cheeks (like a lion's mane), to the gills and below the mouth.
"Goosehead" or "High Cap" typically referrs to unusually tall headgrowth.
Ideal Fin Shape
Fin dimensions can vary widely. These are simply considered the ideal. Normal (common) goldfish have one Dorsal, Anal, and Caudal fin and a pair of Pelvic and Pectoral fins. The chart below shows a Fancy type, with split Anal and Caudal fins:
There are several different standard fin types:
Fantail finnage is the standard among double tailed breeds, and the ideal for Fantail goldfish. It is similar in shape to the Common Goldfish. Fins are short or medium length (1/4 to 1/2 the length of the body), and stiff. The fins should be slightly rounded and the caudal fin should have a mild fork. Common Goldfish have Fantail-type finnage except that they only have a single tail.
Ribbontail finnage is similar to Fantail finnage but longer and less rounded. The Caudal fin should be long (1 1/2 times the length of the body) and flowing...curling in on itself in adult fish. The dorsal should be high and erect, like a Ryukin. True Ribbontail goldfish are actually a variant of Fantail goldfish, but ribbontail-type finnage can be found on other breeds as well.
Fringetail finnage is similar to Ribbontail finnage but shorter. The fins' edges are serrated, unlike the straight edges of the other fin types. The bumps are all uniform and round however, and should not be confused with the erratic edges of straight fins that are shredded or split. The caudal fin should be about 1/2 the length of the body. The dorsal fin should have a standard straight edge to it.
Veiltail (aka "Broadtail") finnage rare. All the fins are very long, with a caudal fin twice the length of the body. All edges are straight, and the caudal should have no indentation. In the Veiltail the finnage is flowing, while in the Broadtail it is stiff.
Butterflytail finnage is long and erect, but the caudal is completely divided and splayed out to the sides like butterfuly wings. The caudal fin is ideally rigid and rounded with a mild fork.
Ideal Color
Goldfish color is stimulated by light, the same way our skin tans in sunlight. If placed in the dark a Goldfish would eventually turn white (disclaimer: never place goldfish in direct sunlight however).
Most exotic goldfish come in many different colors. The original gold color was first observed in the 3rd century, but was probably a far cry from the brilliant orange that is standard today, even on Common feeder goldfish.
White and Red varieties were bred by around 1200 (as was Yellow, though commoners were forbidden to own them as it was considered a royal color), and Calico varieties by 1600. Bronze and Blue varieties were bred by 1900.
To date, goldfish have been bred into white, yellow, black, blue (a sort of transluscent blue/grey that usually only occurs in calico varieties), red, bronze (a sort of dull brown/yellow) and mottled multi-color as well as the standard orange. A dark red/brown, often called "chocolate", seems to be a new color that has been developed. "Mottled" means a combination of multiple colors. "Calico" means Mottled with different scale types as well.
Some metallic Gray/Black fish are also referred to as "blue". But "blue" usually referrs to a translucent grey color that appears in calico fish and other fish with nacreous or matte scale types.
Note that "white" is not the same as "albino". Albino fish have no pigment, so their skin is transparent. White fish have actual white pigment in their skin that makes them white. Albinos can be identified by having transparent skin and red eyes. Non-albino goldfish with white pigment have neither of these traits.
Some varieties will only come in a single color or style (Such as Black Moors and Wakin). Some specialty colors transcend breeds. Specialty varieties include:
Tancho (aka "redcap") - These are completely white (not transparent) fish with red spots on the top of their heads. The coloration is typically associated with Orandas, but can be found on almost any breed of goldfish. The red coloration is, ideally, a perfect circle or oval.
Calico - This is NOT the same as "Mottled". In a Calico, there is usually a predominant color in the background with several other colors (two or more) overlapping it with areas of transparency. Calico fish always have nacreous scales.
Mottled - This describes any combination of two or more colors in random patches on the fish's skin Mottled fish do not have nacreous scales...they are Matte or Metallic. In a mottled fish, the colors do not overlap or bleed into each other. They are distinct patches.
Panda - Almost always associated with Moors/telescopes, a Panda is a black and white goldfish with black eyes and white coloration patterns like a Panda. Though this is the desired coloration, many goldfish are called Pandas simply because they are black and white.
Not all Moors with this coloration are true Pandas...some are merely going through a transitional color change. The black areas fade as the fish gets older.
Twelve Reds - This referrs to a color pattern on a white goldfish with red spots in twelve locations. It is usually associated with Jinkins, but can be found on other varieties as well. It is typically a white fish with red on all fins (or the tips of the fins), the lips, and the gills.
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