Betta Fish Tail Shapes

Plakats
Plakats are a name for short tail bettas. These are often referred to as fighters, but they can be very good pets and good looking fish. Plakats have lately been bred to have fancy colors and bases for fancier finnages such as halfmoon plakats.  Good plakats should have a nice rounded dorsal fin and full tail fins.
Veil Tails (Vt)
Veil Tail bettas are byfar the most common of all of the tail types.  Veil Tails are well known for having the longest finnage of the tail types and can be easily identified by having long curved fin rays.
Double Tails (Dt)
Double tail bettas have been around for some time. Becareful on what you buy.  Some single tail bettas are often sold under twin tails meanwhile they have just fin damage. True double tail bettas have an extended dorsal fin. Basically this means that the dorsal fin start much closer to the head of the fish and you can also notice that the dorsal has twice as many fin rays than a single tail betta has.  Show double tails should have a full split between the two lobes. Bettas with a poor and shallow cut are sometimes called hearttails, but in fact these are poor quality doubetails.  Some varitations of double tails include names such as tripe or quad tails, heart tails, split tails, double split tails etc..
Deltas and Super Deltas
Deltas look like veil tails, but they have perfectly straight tail and fin rays.  This making the tail somewhat shorter.  The key thing are the straight tail rays.  Super deltas look like halfmoon bettas but they lack the tail extending to 180 degrees.  So anything under 180* are considered superdeltas.
Halfmoons and Over Halfmoons (Hm, OMH)
Halfmoon bettas like stated, have a tail that reaches 180 degrees.  Sometimes their tails will exceed this number. These bettas are called OHM or over halfmoons. Halfmoons have taken the market by storm and often referred to as the king of bettas. Good halfmoons should have full finnage and no signs of holes  or cuts in the finnage.  Halfmoons are very uncommon to see in local fish stores, try looking online for good breeders. Some halfmoons have 8-16 rays. This will give the fish extra length in the finnage.
Crowntails (Ct)
Crowntails over the past two years have become very common in local fish stores. Alot of chain stores have started to sell crowntails in massive numbers. Crowntails are bettas that have fin rays that extend past the fins itself. Giving the betta an appearence of raggy or fin damage. Good crowntail bettas should have thick and full rays. When they are thin and brittle it could mean the fish is weak and unstable. Other types of crowntails have appeared. Double ray crowntails are bettas that have the extended rays that branch off into two. Some other common varitations include DDRs, King cross, single rays, double rays, double tail crowntails etc.
Comb Tails
These bettas are similar to crowntails, but their fin rays dont extend as far.  They are like little stubbs at the end of the fins.  They often have straight tail rays and are one of the harder tail types to find.
Round Tails
Round tails are not short finned plakats. They are long fin bettas with like the name says, rounded tail fin.  Not very common, but lovely looking fish.
There are plenty of other tail shapes, but what I listed above are just a few of the common and basic tail shapes.  There are more comming, and soon I will upload photos of different tail shapes.