The Roundtail

The term "roundtail" is most usually used to describe a single tailed betta with rounded caudal edges. Although many of these can look very symmetrical, they would not place well in IBC sanctioned showing, which favors the 180 degree angle and straight edges of the "HM" variety.

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The Doubletail

Blue Doubletail male (DTM)

The Double Tail betta is also known as DT or DTM (doubletail male), and DTF (doubletail female). They are a cultivated mutation that gives the fish a dorsal fin equal in length to the anal fin, and two distinct caudal lobes. Often it also produces a thicker-bodied fish with a wider backside, or deformities such as crooked spines.

Breeding DT into regular ST or HM lines is widely practiced because of the advantages the gene brings, such as wider dorsals and better branching. It is always recommended to cross DT to a single-tailed type only, since DT x DT often produces deformed fish.

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The Spade Tail

Spade Tail red betta

You just don't see Spade Tail bettas very much anymore, though they were more easily found when I first got started in bettas in the early '90s.

This tail form is pretty basic; the caudal has a wide base that narrows to a delicate point, like a spade. Having never bred Spade Tail to Spade Tail, I can't say for certain whether the trait even breeds true. I can only give you the results of my experiments breeding spade to other tail types (like DT or delta) and can report about 30 - 50% spade tails from those spawns.

I think it can be safely said that most "spade tails" are simply a variation of the Veiltail, and pretty commonly seen on females and juvenile VTs whose finnage hasn't reached full weight and length.

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The Veil Tail

White veil tail betta

The most widely-available longfin form in bettas is unarguably the Veil Tail betta (or VT). This is the natural form the fins take when they are artificially lengthened from the wild shortfin form with selective breeding. This is the type of finnage usually seen in Pet Stores. Though attractive in its own right, it is generally considered undesirable among those breeding for show-quality form. In fact, the IBC (International Betta Congress) doesn't even have classes for Veil Tails any longer, and it has become a sentiment of negativity to describe any high-bred betta of 'going veil-ish'.

The Veil Tail betta has a caudal that tends to arch and then slope downward, like a wedding veil. 

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