Factual error: When Arnold's is seen from the inside, metal pinball machines are seen. Since this show is set in the 50's, the pinball machines should be wooden. Metal pinball machines didn't show up until the 70's.
Continuity mistake: Ralph is at Al's after everyone else has left on their dates. He's depressed and as he walks up to a table, you can see a couple of plates of food sitting on it. Then Fonzie comes in (hitting the wall to turn the lights on) and Ralph looks away from the table (he doesn't move otherwise). Then they show the table again - which is completely empty.
Factual error: Joanie, looking at an issue of Billboard magazine, says, "Chachi's record has only been out two weeks and it's already #178 on the chart." Since 1959, Billboard has had a weekly "Bubbling Under" chart in addition to the Hot 100. In the 1960s, when this episode takes place, the Bubbling Under chart listed no more than 35 songs (#101 to #135). There is no way Chachi's record could be #178.
Continuity mistake: There is a blizzard on Christmas Eve dumping two feet of snow on Milwaukee. Mrs. Cunningham, LaurieBeth and Al are stuck at the Cunningham's house. Near the beginning of the episode, they show an exterior shot before they go inside the house. The driveway for the house is wet looking but totally clear of snow, with no one there to clean it, and it's still snowing. The hardware store has two feet of snow outside of it.
Continuity mistake: A pilot takes Chachi up in a small plane so Chachi can go skydiving. In the long shots showing a real airplane in flight, the sky is deep blue and cloudless. In the close-ups, filmed in a studio, the plane flies through several wispy clouds, likely formed by a wind machine blowing the mist rising from dry ice.
Other mistake: The background action in the opening credits is slightly unsynchronized. This is most obvious at the "Goodbye gray sky, hello blue" line in the theme song, as Potsie mouths the lyrics. It's also obvious at the "My cycle hums" line, as the girl in the blue plaid skirt shows up at "cycle", rather than "my." (01:13:10)