Support conversations are major sources of this. Example from The Sacred Stones: When Ross gives Amelia a necklace he'd bought awhile ago from a street beggar despite not having anyone in his life to give it to, her upbringing in poverty has her unwilling to accept something so precious to him. His response to this to say.
Ross: I see. Please just take it and enjoy it. Amelia: Really? I can have it? Ross: Of course! Besides, I've always looked better in earrings than necklaces. Just kidding!
And he's just a minor character shipping with a minor character.
Almost anything regarding Sain would count, really, but his support with Louise definitely takes the cake, for one simple reason: in the C-support, he realizes a moment too late that the lady he's flirting with this time is already married. Whoops. Cue self-scolding and intense inner conflict, made worse if he builds up to a B-support with her (the highest he can get with her, by the way). Meanwhile, Louise's reaction to all this?
For all of its bleak moments, Geneology of the Holy War has a moment that qualifies here: Patty and Shanan's recruit conversation. Shanan goes to grab Balmung from the Shrine, Patty already has it, he has to convince her to give it to him and reveals his identity... and Patty starts fangirling him right there.
Hector's path's version of "Victory or Death". Send out either Farina or Serra to the ruins where Sophia appears all of a sudden. Hilarity will ensue as either girl's dialogue part will almost exclusively made of her screaming her bloody head off.
Fire Emblem: Sword of Seals has the wonderful Saul/Dorothy supports. Especially the C support.
Saul: While I was doing my job, a young maiden was being harassed by a very rude man.
Saul: The man fled with haste. However, the maiden was also gone when I looked back...
Some of Rutger's earlier supports in Sword of Seals tend to be pretty funny. His C support with Dieck gives the latter the idea that Rutger was trying to kill him simply because he was bored. However his declining to spar with Fir in their C support immediately leads Fir to believe that he's sexist is even better. But it's not because he's sexist. He doesn't play around if he has a weapon in his hands, he kills people. Funny and Bad Ass.
His A support with Clarine is pretty funny too, when she tries to be nice to him for once, and Rutger's just like "don't try to be something you're not, you're better at screaming at me".