Fred: If they see us on Peek-A-Boo Camera, we're dead!
Barney: Oh, worse than that, Betty will kill me!
The second credits sequence (which is considered more popular than the first one due to Memetic Mutation). At the end, Fred would put out the cat, which would jump back in through the window and toss Fred out. Fred is forced to bang on the door and shout for "WIIILLLMAAAA!!"
The "Happy Anniversary" scene near the end of the episode "The Hot Piano", where Barney and a group of policemen (It Makes Sense in Context) constantly interrupt Fred with their singing as he tries to give Wilma best wishes on their wedding anniversary, much to Fred's increasing frustration.
An episode had Fred and Barney assisting what turned out to be a female spy. She asks if they're married, and Barney replies "Nope... we're just good friends."
Fred is presented with his godchildren, who he says are all named "Fred." Wilma asks if that would get confusing. Fred says, "Not really, seeing as they're all girls."
In an episode where Stony Curtis comes to the house, Wilma has Betty pretend to be her "upstairs maid".
Fred: We don't have an upstairs maid! WE DON'T EVEN HAVE AN UPSTAIRS!
In "Son of Rockzilla", Fred (stuck in the namesake monster costume) is been chased by Doris the Finkasaurus (resembling the movie monster) that escaped from the zoo, and then the police. They all run over Barney, who came back with pliers to get Fred's costume off.
Barney: Just for that: I won't help you take off your phony head!
"The Long, Long Weekend": A female hotel guest is terrified by the sight of Fred and Barney crashing into her room, and calls the manager. The hotel manager picks up the phone.
Manager: Front desk.
Woman: AAAAAAH!!! (unintelligible sobbing)
Manager: ...Would you repeat that, madam?
Manager: No! Not that part!
Woman: Oh. (unintelligible sobbing)
Fred feigning an ailment in "The Flintstone Flyer" as the two couples are about to go to an opera. Both the English and Spanish dubs of said scene.
"Wilma Flintsone, a natural redhead?"
A classic case of Never My Fault occurs in the episode The Astra'nuts when Betty accidentally gives Wilma the wrong address for Fred and Barney's physical exam, Wilma gives it to Fred, and Fred drives them both to the army recruiting office, resulting in the two being enrolled in such.
Barney: All I know is it's Fred's fault!
Fred: What do you mean my fault?! It's Wilma's fault!
Wilma: It's Betty's fault!
Betty: It's your fault! (The camera pulls back to reveal that Betty is angrily staring at herself in a mirror. She realizes this) Wait a minute...let's do that over again, and I'll start it this time.
The "Dr. Sinister" episode has some great running gags.
The guys constantly attempting to explain they're not spies. "We were on our way to get brontosarus burgers..." "And some buns..."
"Oh, no, Madam Yes!"
Madam Yes exiting the scene calling for Fred and Barney "don't tell them anything." "But we don't know anything!"
When they find themselves locked in a room with Madam Yes, Barney starts pounding on the door and begs Dr. Sinister to save them.
The ending as Fred and Barney find Madam Yes showing up at the door and immediately slam it and start nailing boards over it.
Fred going on a rampage in at Mother's place when he sees Wilma on TV doing her cooking show, while he's stuck chewing on a steak sandwich that resembles the bottom of a shoe. Specifically it's a diner run by a man named Sam Mother. Also, the one-sided conversation Barney has with Fred when he calls from jail that makes it sound like he hit his mother and smashed her television.
Barney: Yeah? Oh, hiya Fred. What's new? What are you doing in jail, Fred? You broke up a television set? In Mother's Place? And hit Mother? Oh. Okay, Fred. I have to go down and bail Fred out, Betty.
Betty: Why you ought to leave him there. Anyone that would strike his mother and break her TV set should be in jail. Those old folks love TV.
Barney: Not Fred's mother, Sam Mother.
Betty: I don't care whose mother it is, it just isn't nice.
Barney: But he runs a beanery, you know a restaurant.
Betty: It makes no difference what he does. We're concerned with his mother.
Barney: But, but, but Sam Mother who owns it... Ohhh skip it. I'll explain later.
Fred going on a diet as part of a game show challenge, where he has to drop 40 pounds, and his increasing desperation for food, including attempting to steal a bone from Dino.
The feud between Fred and Barney when Barney confiscates Freds bowling trophy due to his refusal to pay his membership fee for the gentlemens club they're part of. It quickly devolves into a turf war, with Barney sneaking into Freds yard to take back the various things Fred has borrowed over the years, and then running like hell back to his own yard when Fred catches him.
In "King for a Night", where Fred switches places with a look-a-like King of Stonesylvania, we see multiple times knives being thrown at him from off screen. He jokes that he is thankful there is not one good knife thrower in the entire kingdom.
During the Scooby-Dooby Doors scene, Barney chases after Fred because his look-alike beat him up. Then comes Wilma and Betty, who chase after him thinking he's "mixed up in this too".
Barney:(after getting hit on the head by Betty) Hey Fred, wait for me!
This line in The New Fred and Barney Show episode, "Haunted Inheritance".
It may be seen as sexist now, but the running gag in "The Sweepstakes Ticket" of Wilma and Betty, before going shopping, they do the battle cry of (da-da-da dat da-dah!) "CHAAAAAAAAAAARRRGE it!!!" And later, in the same episode, Fred and Barney do the same cheer before getting new bowling balls.
In "The Picnic", Fred is wrestling with the moral dilemma of dropping Barney as his partner in the Water Buffalo Picnic Games (as they always lose) with the more athletically-competent Joe Rockhead (who always wins). Wilma scolds him, saying that doing that to his best friend over a trophy isn't a nice thing to do. Fred retorts that "Nice Guys Finish Last". That night he has a dream...
Wilma:(in the dream) He's my husband, and I ought to know he's a nice guy.
Fred: Nice guys finish last...
Barney:(in the dream) Fred Flintstone is my best friend. He's the nicest guy I know.
Fred: Nice guys finish last.
Betty:(in the dream) Fred's a very nice guy.
Fred: Nice guys finish last!
Grand Poobah(in the dream) And the booby prize again goes to Flintstone and Rubble, a couple of nice guys who, of course, finished last! (loud, mocking laughter is heard)
In "A Haunted House is Not a Home", Fred's not-all-there uncle, Giggles, has passed away and he leaves his inheritance to his nephew. However, there is a catch: Fred has to spend the night in his uncle's haunted estate. On top of that, if anything were to happen to Fred, the inheritance will be divided up between the uncle's staff, including his butler, his cook and his gardener. As you would expect, in this haunted house, the staff all try to get Fred (and Barney, who is spending the night with him) killed so they can claim the inheritance. But the real kicker is that Uncle Giggles never died at all; he had faked his own death as a prank. Fred's reaction has to be seen to be believed. It doubles as a Moment Of Awesome for Fred, with Jonny Quest-sounding music to boot (that incidentally had just recently debuted at the time of this episode).
The can opening bird saying "This job sucks!" after opening a coke bottle.
It's especially amusing when you bear in mind that the original show's many animal-based "machines" had a penchant for breaking the 4th wall, shrugging at the viewer and justifying their occupation by saying "It's a living". So having one give his full negative opinion on his job (and the fact that slang like that wasn't appropriate or around in the 1960s) is a bit of a humorous switch.
Fred's response to the bowling championship game, and Barney's concern over the close score.
Barney: You need a strike to win. Can you do it? Fred: [confidently] Heh. Is the earth flat?
Sharon Stone: Can I get you anything? Coffee?
Sharon Stone: How would you like it?
Fred: ..In a cup?
Sharon Stone: Bold choice, Mr. Flintstone!
The families came home to see Dino and Pearl tied up. Fred immediately went to untie Dino, with Pearl growling in anger.
Fred tries to think of a way to stop the machine, and he suddenly has a random thought of Pearl getting eaten by a dinosaur.
Betty: That's right. He plays Dr. Gravel on The Young and the Thumbless.
Fred, going door to door and trying to honestly sell vitamin supplements rather than lying or having a "hook," has managed to move exactly one bottle (to a couple of hippies who think they'll get high off it). He figures Barney is having just as bad a time... and then we see that he's selling them by showing Bam-Bam's super strength to a thrilled crowd.
When Professor Sargon is giving a lecture on the the universe, and he says it's hundreds of years old and runs on loneliness.
Joe calling the suicide prevention is grim, but it's funny.
Joe: I think I need help.
Hotline: You've reached the veteran's suicide hotline. Please hold!
Joe: Okay, but this hold music better be pretty damn good.
Issue 3 A Space Oddity has Carl Sargon show the Bedrock Middle Schoolers their newest rocket launch with the chimp, Sergeant Grumbles (who may actually be a sergeant). How do they get him into space? By dropping an Apatosaurus onto a big target and hoping for the best.
Pebbles asks Bamm-Bamm where's he last place he'd go if he had the day off and they go to Bedrock Cave of Science and Technology.