The second credits sequence. 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. The three 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.
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 is a bit of a humorous switch.
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 Wilma's mom tied up. Fred immediately went to untie Dino, with Wilma's mom growling in anger.
Fred tries to think of a way to stop the machine, and he suddenly thought of Wilma's mom getting eaten by a dinosaur.