Any time Mion "trolls" Keiichi, especially when she starts acting like a pervert as part of it is guaranteed to generate some lulz. At its best in the manga.
The parts where Shion trolls either of them are infinitely more hilarious, especially in the Watanagashi arc. Even that is taken up to eleven in the English anime dub, where her voice actor just becomes so blatant about it that you can't help laughing.
The pool episode.
Especially at the end when you realize what will happen if/when one of the men wears the 'magic trunks' for three hours...
Chie-sensei getting angry at Keiichi over curry (nevermind that Rika lied) and whipped out T-squares made this troper laugh when all her Expyness to Ciel was unsubtly lampshaded.
Then there's Rika bribing Hanyuu.
The chorus to Keiichi and Oishii's character rap song, "Uwasa no Jikenbo ABC." Yes, that's right. Rap. Cue the beat:
"Do you know Oyashiro-sama? Yes! Hinamizawa! I kill you, you kill me! No, Hinamizawa!"
From Episode 9 "Why are you holding a golf club?"
Episode three of Kai does this same scene again, except it's even funnier. Same episode includes Keiichi giving a long speech to a baseball player about how there's nothing wrong with being a pervert so the Hinamizawa baseball team can beat his team.
Also, at the beginning of that scene where Mion, in a rage, throws the baseball bat at the ground only to have it bounce back up and hit her in the face.
The VN's version of this is arguably even more hilarious. Mion is beat up for some reason, and when Keiichi asks, she mentioned that she tried a strategy that didn't exactly work, complete with everyone commenting on her bravery for doing such an action. It is never explained what exactly happened.
From Episode 14, Rika asking about Akasaka in the English dub always gets a chuckle out of me.
Mustache Man (If anyone knows this guy, just say so): I heard the guy they sent down here was a newbie. That's about all the information I've got though.
Rika: A newbie? Like a really fresh newbie?
Mustache Man: Yes, definitely new and fresh.
Rika: Yay! A young, fresh newbie!
Mustache Man: Well he certainly seems to be all that.
The last episode of Rei, where Rena accidentally swallows a magatama that will make her fall in love with whomever carries its counterpart. As the magatama ends up in the hands of Tomitake, Takano, and finally Oishii... Hilarity Ensues.
In the same episode, the scene where Rena punches Satoko, sending her tumbling several feet and sliding beside Keiichi and Rena. The sound she makes when she goes flying is hilarious.
Also from the same episode, Mion punching Keiichi repeatedly while Satoko holds him down is rather funny to watch, especially with the expression on Keiichi's face.
And still from that episode, Rena's "USO DA!!!" and subsequent brief Yandere moment. Three times.
Then there is this hilarious exchange between Hanyuu and Rika after Rena confesses to Keiichi.
Hanyuu: Auu~ Rena just confessed her love to Keiichi!
Rika making use of her spiritual connections with Hanyuu by feeding her wine and spicy food to punish her.
The manga has multiple arcs that weren't from the games, that are usually comedy based.
"What is so wrong with guys being perverts?" It's funny because you don't think it's true.
In the first episode of the Tatarigoroshi-hen chapter:
Rika: "If Satoko and I hadn't come along, the flames would be going 'bou, bou' and the fire truck would be going 'woo, woo' right about now."
The point in Meakashi-hen where Mion tells Shion about Keiichi's obliviousness and gets sad...so Shion decides to make her happy by joking about how she'll compete with Mion for Keiichi and working her twin into a rage. Trolling: it's a public service!
Chie-sensei: "Ryuugu-san! Do not suddenly take the transfer student home!"
From the same scene Satoko punching Mion in the stomach to shut her up after she asked the cute new girl what color her underwear is
Again from the same scene, the fact that after the 2nd time Rena tries to take Hanyuu home, they tie her up. Bondage-style. She proceeds to break the ropes and grab Hanyuu a third time.
Where is Miyo Takano's fangirl moment? Anything that shows the usually emotionless Big Bad chibified and sexually rubbing herself against a door definitely counts, she even steals Rika's catchphrase and squeals "NIPAAA~"
This is especially funny considering that earlier in that episode Rena the shows usual fangirl was seen acting the same way (complete with rubbing) to a stool (but that is normal for her).
To add to the funny, don't miss Tomitake's bug-eyed look of WTF in the background when she first starts her fangirl moment, or Hanyuu and Rika muttering to each other about how this is completely ruining their image of the normally sedate Takano.
The scene in Kai of Rika downing Hanyuu by eating spicy food and wine.
Hanyuu's reaction to Takano describing her as a bloodthirsty god that ruled through fear was to cutely and angrily stomp and shout "That's not true! That's not true! I hate you, Takano! I hate you!".
The water gun fight at the beginning of episode 22 and the resulting punishment game, it was especially funny to see Satoko's prank on Rena not stop her from eliminating the Bratty Half-Pint, and Mion being forced to wear the ridiculous outfit (which included a tail) she picked out for Keiichi. Since this happened right after one of the darkest episodes, the Mood Whiplash can not be overstated.
Mion's reaction after her and Shion's Lover Tug-of-War with Keiichi being the rope. She yells at her to stay out of town and tells Rika, who is a priestess, to throw salt at her twin — which is the traditional way of keep demons away from an area.
Nope, only Mion does (and technically, it's only one part of the first character, which means "fascination"). The character they have in common means "sound".
From the Watanagashi-hen visual novel: The club game is a game where they are given a topic and they have to write down the first word that comes to mind. If someone else has the same word as them, they get a point. The theme is "Summer vacation". Everyone writes down things such as "Summer festival" and start getting all excited about the idea of summer vacation while happy music plays. Then Keiichi shows the others his word. What word did Keiichi chose? Homework. Cue the happy music stopping.
Just Keiichi losing at "Sympathy" in general.
From the same arc, Chie forces her students to make curry which they will be graded on. Keiichi, not knowing how to cook, makes... a rice ball. He then precedes to give the teacher a huge speech on how much love and effort he put in on making the rice ball. He gets a perfect score.
Only for Chie-sensei to pull him aside a give him such a curry lecture that afterwards he keeps mumbling "Curry... curry..." over and and over. When Rena sees him, she says his eyes look like curry!
Again from the Watanagashi-hen visual novel, Shion shows up at the Watanagashi Festival and decides to tease her twin by pressing her breasts against Keiichi's arm. Rika sees this and tries the same thing. Too bad Rika doesn't have anything.
The Zombie Tag game in episode 2, especially with how dramatic it is.
Toward the end of the Tsumihoroboshi sound novel, after the hostages have been freed an officer who pulled rank on Ooishi (and barely understands the situation to begin with) begins making stupid orders. Ooishi handles him.
The scene in the second Higurashi Kira OVA when Rika transforms into her magical girl form, only to realize that she's still a child and that her chest is still flat. Sorry, Rika. It looks like you'll never beat Mion.
Keiichi's first attempt at cooking by himself. Seriously, did he use cooking oil or lighter fluid?
Keiichi: I'm like a master chef!
At the All Cast Review Session of Himatsubushi-hen, Keiichi (who at that point still didn't have a picture in the game) takes advantage of the lights being turned off to harass the girls since he didn't have any appearance in the previous All Cast Review Sessions due to not having a picture. Then Chie-sensei channels Ciel and starts attacking Keiichi with Black Keys and the freaking Seventh Holy Scripture. Hilarity Ensues.
In Musubienishi-hen Rika and Satoko spin Keiichi around blindfolded to decide who will be his girlfriend out of Rena, Shion and Mion. He ends up choosing Shion and Mion's grandmother.
Then it cuts to the two of them at a picnic with her feeding him while the others look on in a mixture of shock, disbelief, and horror.
The whole game of rich-man, poor-man in Onikakushi, extra mentions going to the principal's fighting game skills.
In the final Visual Novel, Shion gets one right in the middle of a dramatic You Shall Not Pass scene. Keiichi tries to insist on staying, but tells him to go... because she does not want to have to explain how a Kalashnikov works.