During Otis' sobriety test, Barney wears his cartridge belt upside down. It can be seen clearly when he collapses into the chair.
All the champagne glasses are empty.
When Barney shows Andy the chart he made showing the lineage of Nathan Tibbs, he tells Andy that "Helen married the 2nd cousin of Duncan Phyfe", but her name is shown near the bottom as having married *Duncan Phyfe*, not his 2nd cousin. Furthermore, they are shown to have only 3 children, all daughters (Mary, June and Agnes). If Barney came after any one of them, he couldn't be a Fife (spelling it correctly, as he says) because their names would have been changed by marriage.
As Andy and Opie are getting ready to walk into the courthouse, they stop to talk to Jud and another man sitting on the bench. When they finally go into the courthouse, Andy goes to close the door behind him and a crew member's hand is visible, grabbing the outside handle to help close it.
Although Mayberry is a "dry" town in a "dry" county, Mayor Pike serves champagne after the presentation of Otis' plaque.
Andy states that Barney's middle name is Oliver, but in a different episode, when they are looking at their pictures in a high school year book, his middle name is Milton.