One of the biggest is that Dungeons and Dragons leads to Satanism. Fascinating article I dug up on Wikipedia after reading about Gary Gygax. Ms Patricia Pulling was a troubled individual. Her son committed suicide by shooting himself, and she goes off on a crusade to sue TSR Inc. and the school principal due to some connection between her son's death and Dungeons and Dragons. She formed the organization Bothered About Dungeons and Dragons, and went along with her crusade against the occult.
My own mother was quite convinced. However, evidence shows otherwise, and the are much worse vices that cause far greater harm to youth than RPG's and 'the occult'. Her cases all lost in court. To include some where she was an 'expert advisor' on. Why did they lose? Because she had little working knowledge of the occult, and in her book "The Devil's Web: Who is Stalking Your Children for Satan", she actually encourages Law Enforcement to question suspected teen occultists to ask "Have you read the Necronomicon, or are you familiar with it?" For anyone who may not know (don't be embarrassed, I didn't know until I was 20 myself...) the Necronomicon only exists in the fiction of H.P. Lovecraft and other fantasy/horror authors (or movies like "Bruce Campbell Vs. Army of Darkness"!). In 1990, Michael A. Stackpole went through and de-bunked a lot of the myths in the "Pulling Report".
If anything, RPG's are as benign as any other media that an individual might get 'hooked' on. Just look around and you'll see what I mean. Sports bars are a fine example... Sit around a TV, get drunk, obsess about a sport, and then riot when the ticker tape parade happens (Los Angeles!).