When Heiji confronts Conan on being Shinichi. In some dubs, he even takes time to call him out on how shitty his accent is (Or how badly he imitated his voice). In the original, he speaks using Conan's bad attempt at his accent upon 'awakening' from the tranquilizer, which gives Conan a false sense of reassurance. Made even more hilarious because Heiji only woke up from being anesthetized due to Kogoro hitting him upside the head, which causes Conan a borderline Freak Out.
Also any time Heiji slips out Kudo when others are around and them trying to cover it up.
Any time Conan makes a sarcastic comment to himself.
Any time Sonoko tries to help Ran in her "relationship" with Shinichi and Conan starts thinking perverted thoughts.
Especially when Sonoko takes a photo of Ran in her new swimsuit and sends it Conan calling it her "Special Sexy Shot of Ran". Then Ayumi finds him on the phone and Conan nearly gets his cover blown. Ran asks Shinichi to delete the photo, but since he already saw it, he saved it. Then the Detective Boys found it and had Ai delete it. Conan's reaction: priceless.
Any time Kogoro goes into his "protective dad" mode over something trivial.
Like when Ran gets a call from Shinichi and Kogoro starts eavesdropping.
Any episode featuring Ran using karate on the culprit.
Also, the scene featuring the Sickeningly Sweethearts Noriko and Yukio, cooing and saying cutesy things to each other while Kogoro is all "please stop".
Nobody in Tsukikage Island recognizes Kogoro Mouri and they even mistake him for an astronaut, making Mouri sulk in the anime.
In Episode 170, Dr. Araide (Sr.) tells Kogoro to be careful about his health because he has the same name as a famous detective. Kogoro breaks the news to him that they are in fact one and the same.
Another funny line in the North American dub:
"What happened? I heard a woman scream?!"
"Stop talking! Everywhere you or that kid go a corpse turns up! Come on man! Haven't you ever notice that?!"
When Conan unmasks Magic Kaito in his first DC appearance. Kaitou is dressed as Ran. Wearing her red dress. And he actually tells Conan that Ran must be all cold and naked wherever he hid her.Conan'sreaction is priceless.
In Episode 96, when Ran is this close to discovering Shinichi's secret... Yukiko shows up. Holy Mood Whiplash, Batman, it goes from a Tear Jerker to comedy gold in less than ten seconds.
A very good one comes from the first season. Ayumi is stuck in the trunk of a car with the head of a murdered little girl, and the people in the car are the murderers, so Conan and the Detective Boys are following them. It's all very tense until... it turns out that the "murderers" are actors, the head is a prop, and their suspicious conversation was actually their rehearsal for their upcoming show. Made even better by the Cool and Unusual Punishment they get: one of the actors is out cold so the kids have to use a huge longcoat and a wig to replace him, and Ayumi (the one who started it all) has to act as the dead little girl.
Her line during the last scene in the dub is also good: "I am never playing Hide and Seek again!"
In Chapter 400, Haibara gives Ayumi permission to use her first name, which Ayumi has spent the whole case trying to do. The funny comes in when the boys ask if they can use Haibara's first name too, and she shoots them down before they can even finish the question.
Don't forget Episode 16, where the killer starts to attack the sleeping Kogoro who suddenly stops midway through his deduction. Conan's face is so funny.
Even more when Kogoro wakes up and starts running around like crazy, asking for help, after seeing some blood from the wound the sword marked on his head.
In Episode 625, Ran goes to a school festival and decides to go into the haunted house. The entire time she's so terrified that she carries Conan and squeezes him whenever someone jumps out.
Finally, Conan learns that the Hello, Attorney! he's been working with in the episode's murder was Eri Kisaki, Ran's mother, whom he didn't recognize until then because it had been ten years since he last saw her. He then lets out an epic Skyward Scream of pure, unadulterated horror.
The only scene from the Dub I've ever seen:
Conan: Note to self: Stay away from the psycho with the dead mommy laptop.
During the Furinkazan Case, Conan manages to steal the spotlight by defeating Tatsue with a ball was pretty funny. Given her crimes (five continuous murders and gambling), it was a pretty satisfying end indeed.
The entire last page of the Eye for an Eye case. Kazuha makes a failed attempt to confess to Heiji, who ran off in the middle of her confession to confront Conan after finding out that he confessed to Ran. Heiji's face is absolutely priceless. Kazuha's thoughts are even more hilarious though:
Chapter 830: Kaito KID disguises himself as Sera in his latest appearance. Conan explains how he figured out who he was, and Kaito expresses shock that Sera's a girl. Conan then says he'll look over the incident for today since KID helped him out before. "I will, at least..." Cue this page.
In Chapter 873, while inspecting a body in a bath, Conan accidentally sees Ran half-naked and the resulting nosebleed is featured in one panel as a Gory Discretion Shot. Conan ends up being unable to talk properly for the rest of the chapter because he needs cotton balls to stop the bleeding. He claims that the nosebleed was caused by surprise over seeing the dead body, but Sonoko has her doubts.
Some of the OVAs are hilarious! The first one is a Kaitou Kid/Detective Conan/Yaiba crossover featuring some very bizarre transformations and is of course All Just a Dream, while another has a criminal sounding just like Hattori Heiji without the accent which leads to Conan training Hattori to lose his accent so they could record him for a criminal identification program before they both remember that Conan has a voice changing bow-tie, and yet another episode has Sonoko trying to write a detective play and the Detective Boys giving her various suggestions for the solution with the story teller as the main hero.
Chapter 710 had Yokomizo inspecting Eri whom he suspects as a culprit in the Mystery of the Week when Kogoro informs him that he strip-searched her to Yokomizo's shock since he had no idea whey were married until Kogoro told him. What follows is an investigation laced with heavy doses of Like an Old Married Couple routine.
In the fifth movie "Countdown to Heaven" Haibara stops Agasa from eating fattening food at a banquet and leaves to find him something healthier. Heartwarming, but the conversation Kogoro and Agasa have afterwards is pretty funny.