I'm not sure the "fake moon landing" theory is a conspiracy. I couldn't figure out why NASA "conspired" to fake the moon landing. But you gotta admit if they 'faked it', they did a good job .
That's probably a conspiracy that came about because of the speculation that Yuri Gagarin wasn't the first Russian to orbit the earth. Proponents of the Lost Cosmonauts theory argue that the Soviet Union attempted to launch human spaceflights before Yuri Gagarin's first spaceflight, and that cosmonauts onboard died in those attempts.
That was a hot potato in the 60's following Gagarin's historic flight. The secrecy of the then Soviet Union didn't help either.
One conspiracy in the UK that surfaces every now and again is that Shakespeare wasn't the author of the plays and sonnets accredited to him. That all started about 250 years ago when one, James Wilmot a reverend and a literary scholar, set out to write a comprehensive biography of Shakespeare, back in 1781. Wilmot could find no paper trail, no notes, scribblings or papers, so he concluded that Shakespeare's work was that of Francis Bacon. "Where are the books? You can't be a scholar of that depth and not have any books in your home," Wilmot argues. "He never had any correspondence with his contemporaries, he never was shown to be present at any major event, the coronation of James or any of that stuff. I think the evidence that he was not the author is beyond a reasonable doubt."
The conspiracy has rolled on ever since. Over time, more and more celebrities have flocked to the anti-Stratfordian theory: Freud, Whitman, Malcolm X, Helen Keller, Orson Welles, Sir Derek Jacobi. And it’s not hard to understand why: It’s a romantic, glamorous, exciting theory filled with secret conspiracies.
What academics argue is that Shakespeare was no scholar, but if you run an analogy, for example, Sean Connery. Connery was working as a delivery boy at the age of nine, left school at thirteen, had no higher education and became one of the greatest actors in Hollywood. Will people be saying, in centuries from now, "Yeah right?"