Generally, conspiracy theories can be fun--they challenge an otherwise accepted fact, and as usual, the theorist will have an answer for anything. How do you explain the footage of the moon landing if the landing was faked? Soundstage. How did so many NASA engineers agree to the coverup? Payoffs. And why? Because the government needed a morale boost during Vietnam to show we're winning the Cold War.
Of course, some conspiracy theories border on the offensive, such as Holocaust denial (the Nazi one, I mean), or 9/11 truthers (these also lack adequate justification for WHY these conspiracies would take place--what would the masterminds of staging 9/11 gain that they couldn't have done with a much less risky scheme?). But then every now and again comes a conspiracy theory that is generally lighthearted. I'm talking about the long-running "Paul McCartney died in 1966" theory.
In a nutshell--during the Beatles mid-'60s heyday, Paul McCartney died in a car crash, and the British government was distraught due to the immense amount of tax revenue the Beatlemania industry was bringing in. They paid off everyone involved in the conspiracy--producer George Martin, manager Brian Epstein, the other three members of the Fab Four, Paul's family--and hired a convincing look-alike to fill in for Paul. The band survived a few more successful years before splitting into successful solo acts. The fact that they had pretty much stopped touring in 1966 lends credence to the theory.
And the clues--oh, the clues! In the link above, it points to several hints on album covers and in songs that the other members of the Beatles put out various clues that Paul had died. Perhaps the most well-known of the clues is the "Abbey Road" album cover--the Four respresenting key parts of a McCartney funeral: John dressed as a preacher, Ringo as undertaker, George as gravedigger, Paul in formal suit and shoeless as corpse (apparently corpses were buried barefoot?) and the car license plate reading "28 IF" meaning Paul would be 28 then if he were alive. It takes a sinister turn when the theory supposes that John Lennon was killed in 1980 by a man brainwashed by British intelligence because John was going to reveal the truth.
However, the theory has some holes--why, if this were such an important, airtight conspiracy, would the Beatles risk giving it away by sending out clues? Surely if some fans could pick up the clues, there was a good risk that vigilant MI5 agents would as well--especially since the agents were looking specifically for any leaks. Also, wouldn't the British government know there's always a chance Beatlemania could fade, and the death of a beloved performer could if anything lead to more tribute concerts and album sales? If the other Beatles could go on recording with an imposter, surely they could go on as a trio (or with a new member). Also, to find a person who could look, act and SOUND exactly like Paul McCartney? Sort of improbable, particularly when fronting a band as well known and followed as the Beatles at the time. The "new" Paul would not just be filling in, after all, but performing key hits for the band for the next three years and embarking on a successful solo career.
So what of the clues? I think if there's any conspiracy, this was a conspiracy by the band or its management to give out a hint that there was a conspiracy, partly to have fun with their fans and partly to keep stoking public interest in the band. After all, the "Paul is Dead" theory has been a key part of Beatles lore since the late '60s and a playful joke with their most dilgent fans is certainly in the character of the Fab Four.