I'm a huge fan of popular conspiracy theories, like the faked moon landing, 9/11 or Nazis living in Antarctica. Is there anywhere a museum or something similar about conspiracies?
It may not be the the exact controversies you are seeking, but Washington DC's spy museum offers an immersive experience in the art of spycraft.
It is specific to one type of conspiracy, but the UFO Museum in Roswell, New Mexico is interesting.
I will warn you though, the museum isn't all that fancy. Much of it looks like a high school science fair. They clearly aren't a big budget operation, but it is worth the cost of admission at least.
Also about one particular event, the JFK Sixth Floor Museum in Dallas does a very good job covering the conspiracies about JFK's assassination.