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 the episode where Fred is stuck in his monster costume, he's been chased by a female version of the monster that escaped from the zoo, and then the police. The three run over Barney, who came back with a tool to get Fred's costume off.
Barney: Just for that: I won't take off your phony head!
The can opening bird saying "This job sucks!" after opening a coke bottle.
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.