This is a first pass. I am missing decks that are not really an archetype, such as TeleDAD, and archetypes not in the core sets, such as, recently, Weather Painters. I am noting here the first time an archetype had a viable deck. Some archetypes are a stretch, like Crashbugs, and others had a card before there was an archetype, such as Batteryman. It's an art more than a science. Overall, we have approximately 180 archetypes during the first five eras of Yugioh. When we add decks that are not archetypes-e.g. competitive decks, tribal decks, attribute decks; I am sure we can reach 200.
Zexal has more archetypes than the previous three eras.
Arc-V sure has many archetypes. Duelist Alliance has the most archetypes of any Yugioh set to date (excluding reprints such as the Mega Packs, of course).