Well a mutation occured around 150 years ago (possibly longer) that drastically lengthened the fins of bettas. The veiltail was formed. Then came the round tail...I'm unsure of the origin. The delta and halfmoon came from a veiltail betta known as Mr. Great. He was spawned to many females and pretty much traveled the world. Soon more fish with 180 degrees were popping up. Creating the halfmoon. More on this here: http://bettysplendens.com/articles/p...articleid=1024
The starter of the HM Mr. Great:
CTs came from comb tails with less webbing. Generations down the road CTs with amazing extensions and reduced webbing were created. It's one of the newest developments.