For me, it would without a doubt have to be the 'moon landing hoax'. Personal experience has shown me that a massive majority of the American public knows little-to-nothing about the incredible endeavors that took place during the space race. I recently had a discussion with three friends, and I was astonished to find out that they all believe the moon landings never happened. This topic has never come up between the four of us before. I was amazed that they were spewing things about 'no stars', 'intersecting shadow paths', etc. Arguments that are so easily debunked that I'm surprised they're still the number one choice articles of the conspiracy claim. With nothing more than a camera, a few rocks, and 4:00pm sunlight I was able to show how these claims are ridiculous, based on nothing, and take little effort to toss aside as complete rubbish.
Upon further discussion I came to find out that they all only knew about Apollo 11. They literally, all kidding aside, thought that only two men had ever stepped foot on our lunar neighbor. This complete lack of historical education is the main reason that ignorance seems to triumph over reason, evidence, and the beauty of human accomplishment.
This is a conspiracy theory that sadly will never go away. The LRO (Lunar Reconnaissance Orbiter) was able to take many aerial pictures of the landing sights, still in seemingly pristine condition, astronaut footprints still visible in the undisturbed ancient lunar soil. These images won't sway the hoaxers, they seem to believe that an entire new generation at NASA is still covering up a 40 year old government head-game. If you haven't seen these images, I highly suggest taking a gander.
If there's one conspiracy theory that must die, it is this one. The accomplishment is single-handedly mankind's greatest achievement. Our moon has always been a beacon of wonder in our night sky. Our ancestors would never have dreamed that one day a fellow human being would be dancing upon the rim of it's craters. Long live Apollo!
Billy Ray Cyrus sold the soul of his firstborn child (Miley) to the Devil (Disney) so his song Achey Breakey Heart could be a hit, pretty badass conspiracy theory if you ask me