Shermie vs. Bryan's special intro in KOF'98 Ultimate Match. He keeps shouting "WOO! WOO!". This annoys her so she walks over and SLAPS THE TASTE OUT OF HIS MOUTH.
Billy's special opening against Iori is understandable, given what happened at the end of '95. What's less expected is Eiji Kisaragi teleporting in to shake his fist at Iori and then teleporting away before he's spotted.
Yashiro would rib on someone who's got business with the Orochi bloodline outside his own team, by using Chris' "Oh no... he looks strong..." (and ending with "...NOT!"). So how was this funny? Well, switch him to the more monstrous Orochi Yashiro, and he'll do the same while still in the crueler form (the same intro is used in KOF 2002 and its Unlimited Match release, albeit with Yashiro saying Shermie's "I won't like it if it hurts..." (and still ending with "...NOT!")).
Having Team Kim fight against Team Yagami in KOF XIII Story Mode, in which Kim tried inviting Iori to join his Tae Kwon Do dojo, and even prepared a taekwondo gi just for him. And if not enough, he also invited Hwa Jai and Raiden to the sport as well (He offers the Chang-sized gi for Raiden). Hilarity at its finest.
Vanessa: This is Vanessa. I'm investigating the area around the stadium...but there's too much ground to cover. I haven't found anything. I need something specific to go on. I can't just search an area this big for anything "out of the ordinary." I feel like a lost tourist.
Mary: This is Mary. I'm in the slums, dealing with an...unsavory individual. (Screenshot reveals this to be Duck King) I haven't come across anything of note.
XIII Team Fatal Fury Arcade End: During the victory celebration, both the Bogard Brothers are congratulated by Mai (who embraces Andy) followed by Mary with Anton (who approach Terry). As Joe takes note of both of them being fawned over by girls, much to his jealousy, he is seemingly embraced by a woman from out of nowhere. He cheers initially until he asks her who 'she' is only to discover it to be the same transgender man he saved in 2003 who is unfortunately looks nothing like a woman with the stubble, despite the makeup. Cue horrified Joe trying to run off, got caught and end up looking like he's Deader Than Dead...
The Art of Fighting Team ending has Yuri being asked by an interviewer about what would be the ideal husband. Behind her, Robert is holding a ring box, working up the nerve to ask for her hand in marriage. Takuma notices this and stomps Robert's foot before he can say anything. Robert responds in kind and they go on stepping on each other for the rest of the scene. Meanwhile, Yuri is trying to pair up King with her brother Ryo, who seems oblivious to both this and the squabbling behind them.
Chin and/or Kim get Kyo all pissed off via asking him if he actually did finish higschool or not;
It's even funnier if the player chooses Team Japan for Story Mode and finds the Psycho Soldiers. Athena naively asks Kyo if he did graduate or not when he asks about her uniform, Kyo is at a loss of words despite Benimaru telling him it's fine, Kensou is all "WTF?!"...
While the first is removed in XIII Climax, the latter is even funnier. In the original, when Athena asks why would he keep such data, Maxima calmly says he's just interested in her Action Girl strength. In the new version, it becomes a Suspiciously Specific Denial in which Maxima totally didn't store pictures of her for non-fighting purposes. (Hint: yes, hedid.)
King asks Andy to pay for Terry and Ryo's tab at her bar, which makes Andy really damn angry.
Team Kim's ending: Kim not so subtle attempts to ditch his master Gang Il and his girlfriend Luong are hilarious, even more so since they end having the opposite effect.
Team Mexico's ending: Angel and Ramon patting King of Dinosaurs as if he were an overgrown puppy. For bonus points, King of Dinosaurs and Angel even purr Also, King of Dinosaurs/Tizoc admitting he skips leg day every now and then
The special opening between King of Dinosaurs and Mian, where he mistakes her for a fellow Luchador due to her Bian Lian mask. She's completely confused and concludes that he's some sort of lunatic.
Newcomer Kukri spends most of his time trolling every other character in the game.
Moments from other KOF related media (mangas, CD dramas, Chinese manhuas, etc.)
From the KOF: KYO manga's third and last volume, Vice and Mature... re-arranging Iori's tiny apartment to make it cozier. Iori simply can't believe it.