In one episode, the Cunninghams are on their way to the movies. When Ralph asks them what they are going to see Joanie tells him they are going to see Psycho. However, the episode is set in 1957, three years before the film premiered.
Despite more than 50% of the American public smoking during that era, none of the primary cast smoked regularly. There was an episode where Joanie (Erin Moran) did take up smoking;however she was discouraged from this by her parents and friends, which would have been the opposite reaction that people during the series' setting eras would have had.
Frequent Anachronisms. Throughout the run, in the seasons that are set in a short time (like one year), give mentions/depictions of various events of the 1950s (or 1960s) that in reality happened years apart, not close together. (This is largely because the writers of the show did not do their research, to find out what exact events happened in the years they're set in. Instead, they 'threw in' various '50s 'highlights'.)
Many of social stances and character beliefs expressed on the program represent the 1970s/1980s filming dates of the series, rather than its 1950s/1960s setting of the series. This includes a noticeable lack of racial prejudice in later episodes, women being viewed as equals and no one expressing anti-gay sentiments.
Despite several male characters being of the age where marriage was common, if not expected, they remained unmarried during the run of the series. The Fonz, Chachi, Ralph Malph and Potsie Webber were all old enough to be married (and would have been expected to BE married during the 1950s/1960s) yet they remained unmarried during the run of the series.
The changing of the interior of the house after the second season messed up the scheme of the house. From exterior shots the driveway/garage of the Cunningham house was to the right of the front door, while from the interior sets the garage was still to the right of the front door (i.e. it would be on the left as viewed from outside).
The restrooms at Arnold's restaurant have the Men's room on the left, and the Ladies' room on the right. But, in all of the shots of the guys coming into the restroom, they walk in making an immediate right.
In the early seasons of the series, it's clear that many of the actors/actresses portraying teenagers are actually in their 20s+. This is done as to not interfere with the schooling of real teenagers and to give the general viewing audience who are teenagers themselves someone they can look up to.
All though Postie, Ralph and the Cunningham Family are close to Fonzie, with him being friends with Potsie, Ralph and Richie, a real Middle Class family, in the 1950s, would not have associated themselves with hoodlums.