Captain Harriman welcoming Kirk, Chekhov, and Scotty aboard. You can feel the nervous energy surging off him as he greets them the same way as any convention fanboy. You can see that he's visibly repressing a Squee!.
In the holodeck scene, Geordi holds an old-fashioned telescope up his VISOR. His VISOR which, among its many functions, has telescopic vision. Even funnier, he holds it up to the center of the VISOR, above the bridge of his nose.
As the Enterprise is about to crash onto the planet, Data says what everyone on the bridge is thinking: "Oh,shit!"
Pretty much everything funny!Data does that isn't unsettling is this.
(Opening a door with magnetism, in a deep, dramatic voice) Data: Opeeeen Sesame! *Door opens*
Data: You might say I have a magnetic personality. Hahahahahahahaha..... HUMOR! I LOVE IT!!!
Not just his song, but making music with the LCARS touch display at his station. The stares from the crew are priceless.
Riker's "...What the hell?" face at the end.
Not just him, everyone on the bridge!
Deanna actually walks over to look as if her brain is saying "That can't possibly be Data doing that... oh my God it's Data doing that."
Really, anything Data does after installing his emotion chip.
Guinan: Well, it looks like he hates it.
Data: Yes. That is it. I hate this!
LaForge: Data? I think the chip is working.
Data: (drinks rest of beverage, gulps) Oh, yes! I hate this! It is revolting!
Data: *big grin* Please!
Data laughing over a joke that LaForge told during the Farpoint mission.
LaForge: Farpoint? Data, that was seven years ago. Data: I know! I just got it!
Worf's promotion ceremony conducted on an 18th century vessel in the holodeck. After Worf successfully grabs the hat without falling off the plank, Riker simply has the computer remove it from under him.
Riker: Computer, remove the plank!
Picard:Number One, that's retract the plank, not remove the plank.
Riker: Oh. (leans over side to look at Worf) Sorry! (Worf is growling and flailing in the water).
Particularly the look on Worf's face at the moment it vanishes - some combination of Flat "What." and Oh Crap!, and it's on the face of The Stoic.
When Data doesn't see what's funny about dropping someone into cold water, Crusher simply says he just has to get into the spirit. Data thinks he understands... and pushes her overboard.
Somehow, the Dude, Not Funny! reaction from the rest of the crew just makes it even funnier.