All the times the male characters wished they could be as good with the ladies as Rock Hudson... Rock Hudson was later revealed as a homosexual. In one episode, Barney says that he learned how to sweet talk ladies by watching "all those Rock Hudson movies".
In the episode "Man In a Hurry", the titular character happened to be named Malcolm Tucker.
Idiot Plot: Numerous episodes would have been resolved in under a minute had someone decided to tell a slightly uncomfortable truth rather than try to spare someone's feelings. "Dinner at Eight" is probably the worst offender, when Andy was 'forced' to eat three spaghetti dinners in one night and be blamed for being late to one of them, when all he had to do was admit he didn't get a telephone message. Or call when he did get the message and say he was going to be late. Or if the host had called and made sure he had gotten the message. Or the host called when he didn't arrive and ask what had delayed him.
Replacement Scrappy: Warren Ferguson. The guy didn't have an ounce of what made Barney so funny. The fact that he tried to fill Barney's role, but ultimately failed, is what made his character particularly unfunny.
Seasonal Rot: A pretty serious case. After Don Knotts dropped out of the show after the fifth season to pursue a film career, the series went on without him for three more years. Most fans dislike these episodes (easily recognized because they were shot in color) for being unfunny and moralistic. Without Barney to play off of, Andy loses much of what made his own character funny, and he becomes a rather solemn grump.
Also, the sudden addition of new neighbors in Mayberry, such as Howard Sprague and Emmett Clark, really didn't help matters much either.
Tear Jerker: Opie's horror when he realizes that he accidentally killed a bird with his new slingshot in "Opie The Birdman".
The episode "Alcohol and Old Lace" involves two sisters selling moonshine and justifying their crime because "we're not like those other moonshiners, they sell liquor for drinking purposes; our elixir is for special occasions." Naturally, the town's drinkers are making up excuses to take a nip. One particular scene involves a man pretending to be a Muslim celebrating Mohammed's birthday. It's not necessarily a scene that couldn't work today, but it's hard to imagine any show after The War on Terror dropping such a scene in so casually.
A clip of Andy erasing a recording Opie made of a conversation of a suspect and his lawyer made in secret while defending the right to privacy and attorney-client privilege has been making the rounds on YouTube ever since the revelations of the NSA's spying program.
There's also a clip from the end of "Opie and the Spoiled Kid" involving the titular spoiled kid raising a fuss after Andy confiscated his bike for riding on the sidewalk. His temper tantrum gets more and more out of control, culminating in the boy's father promising to sell the bike and implying the boy is going to get a spanking. This clip has also been passed around a lot on YouTube and social media among proponents of spanking.
Wretched Hive: Raleigh. Bit-part bad guys hail from there more often than not, and it is often used as the foil to Mayberry's wholesomeness. Nonetheless, the show is as beloved in Raleigh as it is anywhere else in North Carolina.