Lots of the funniest moments are split up on Youtube for your convenience.
In one bizarre scene, after the girls discuss their suspicions that Makoto is going on a date, Usagi pulls out an idol group's photo book of Rei's and says, "But you've got this too, huh?" Rei's angry reaction is priceless.
Serena running into Jadeite in the episode "Cruise Blues". Serena snuck into the cruise ship in the disguise of a photographer and Jadeite was currently disguised as the ships captain. Jadeite catches Serena in the hold and the pair start to chat, much to poor Jadeite's confusion.
Serena: I got separated from my friends and I took a wrong turn.
Jadeite: I guess that makes sense. There is a big crowd on board.
Serena: (sporting puppy-dog eyes) Oh please, will you help me look for them?
Jadeite: (looking really awkward) Yes, I will. *wait a minute, why am I being so nice?*
The Entirety of Episode 78. (English Episode 71). To explain it, all of the Senshi are down with a virus except for Minako, so she tries to treat them. It goes... badly.
When the girls argue over who should play Snow White in the school play, Makoto claims it should be her because her bust is the biggest. Rei's face is hilarious when Makoto gives her "Want to compare? *wink*" and throws her bust in her face. This is even better in the North American dub, which attempted to censor it by making her claim she had the "most acting experience and talent"... without taking out the BOING sound effect, camera panning down to her chest, the "WTF" look on Rei's face, or the gesture that made it very clear what sort of "talent" she had the most of. Memetic Mutation ensued.
Episode 104, The Shin Chan parody, no words I can put together can say how funny this is it, what's more is that those are the original Japanese voices. 
Even her first appearance as Chibi Moon prompts Moon and Mars to stare in utter disbelief as Moon croaks out a, "Chibi-Usa..."
In a filler episode Usagi was going to meet with some foreign exchange students and her Gratuitous English after getting drunk must be heard to be believed.
"Nice to meet you! Cream stew!" or "Nice tawh meetchoo, cream-oo seetchoooo."
It was also hilarious when Mamoru told the girls about meeting the students, and Usagi couldn't decide whether to feel excited about going to a fancy party with Mamoru or feel nervous about all the foreign conversations awaiting them.
Sailor Moon uses the Moon Tiara Action... With a frozen pizza, as her outfit didn't have a tiara anymore!
Haruka and Michiru are completelytaken by surprise (Luminescent Blush included) as Aluminium Siren whines about how they're standing atop the dining table; and when Eternal Sailor Moon's wings start knocking off things as she walks down the hall, forcing her to lean against the wall. While she makes her usual speech, you can hear dishes breaking in the background. Nothing like the show when it starts parodying itself. Then, when Sailor Star Maker tries to attack, everybody grabs her to stop her from doing so, completely ignoring the Monster of the Week, who receives an Offhand Backhand from Fighter when he tries to get her attention.
The battle against the old art collector in episode 186 is hilarious, specially when he tries to get a power up... and his pose is a Shout-Out to Chibi-Usa's usual invocation of Pegasus in the much maligned Super S season. Refuge in Audacity at its finest!
From Super S: "I am a spider, I can climb...I am a spider, I can climb..." And all of her other antics as Usagi the Red Ninja, Protector of Love.
From the S series, Usagi's secret looks to be about to be revealed to the bad guys and the Outer Scouts when the other Scouts arrive... with Sailor Venus masquerading as Sailor Moon, down to the hair buns.
Doubles as Mythology Gag, as, in both manga and live action, Minako had faked to be the Moon Princess for a while.
From the dub: "And remember, I'm Sailor Moon!"
Gets funnier when you realize the implications of Sailor Venus explains Usagi's Cosmic Heart Compact as merchandise.
Minako was always a little too happy to learn that she really did have a pure heart.
And Eudial and Usagi's reactions when she didn't fall in exhaustion but laughed in happiness. Eudial was frozen in shock, and took a few seconds to give chase when Minako ran away. Usagi? Dull Surprise and a Not Again face.
The same episode reveal the existence of a series of five books titled "Tarisman", with the authors listed as "Haruka & Michiru".
At the start of the episode, Tomoe and Eudial were talking at the phone when Tomoe, ranting about the need to recover the Grail before the Sailor Senshi, accidentally broke the speaker. And Eudial informed him the reparation's cost would be taken by his pay.
Minako moping about it, and her terrifying Death Glare when Artemis dares to suggest she doesn't have a pure heart.
Minako is working her frustration with volleyball... And then realizes she hit a visiting Usagi.
Minako is frustrated by both the lack of attacks and her problems at the crane game when Usagi realizes Minako wants to bait the Daimons. Cue Usagi grabbing Minako's shoulders and shooking her while begging to not do that... And accidentally capturing all the dolls in the crane game.
Luna explains that the Daimons could have simply got it wrong, and brings as example that Usagi, already targeted by them, purely loved Mamoru, purely hates homework, is purely lazy, is purely glutton, and thus was thought to have a Pure Heart.
In the end Minako turns and, with a serious face and a subdued tone, asks Usagi if she wants some of the dolls from the crane game.
Minako meets Haruka and Michiru in the same bookshop with the "Tarisman" books, and they suggest her she could show her pure heart by helping people, and maybe give blood. She runs to do it, and the girls praise her pure heart... Cue Minako running back and giving them a doll from the crane game each.
After the Daimon was defeated and Minako re-hearted, Eudial was about to roast the entire Sailor team plus Uranus and Neptune... Cue Minako hitting her with the dolls from the crane game.
Eudial's reaction at Minako's transformation in Sailor Venus? "You too?!"
Anytime Usagi and Rei argue.
When Rei went to take a bath and saw Mamoru naked in Episode 136.
Usagi spanking Chibi-Usa, but that's not the funny part. No, the funny part was when Chibi-Usa, later in the episode, had a daydream where the situations were reversed and she was spanking Usagi.
Minako dramatically lamenting that "Surely it is a curse... to be THIS BEAUTIFUL!" as the other girls roll their eyes at the nonsense.
Even more so later in episode 141 when the "hot guys" she had been "cheating" with reveal themselves to be Hawk Eye and Tiger Eye:
Later Sailor Venus shows up pissed at them and opens a can of whoopass on them after her In The Name Of Venus Speech mentioned them breaking her heart. They and their Lemures are frantically dodging while wondering out loud what the hell she's talking about.
When Tiger's Eye and Hawk's Eye looked in Minako's Dream Mirror: not only their reaction about her 'amazing dream' implies her life dream has a pornographic nature, but, differently from anyone else, she was apparently excited by the experience.
In the dub anytime Darien refers to Serena's "meatball head" because of her hairstyle.
In both the dub and the sub, Usagi and Mamoru finding out that Chibiusa is their daughter.
From the sub
Usagi: What?! Me a parent?! Oh no wonder the little brat annoys me so much! She takes after YOU!
Usagi's Epic Fail at training to be a proper princess/lady in the episode "Aim to be a Princess? Usagi's Weird Training". The crowner has got to be when the rest of the girls crash the dance class and everyone else passes - except for Usagi, Rei and Minako.
Kunzite and Tuxedo Mask are arguing about whether or not Sailor Moon should be killed or the silver crystal taken. The youma sighs and says she has a headache. And Sailor Moon offers her a cup of tea.
In episode 110, Eudial speaks on Haruka and Michiru's answering machine. The message ends mid-sentence, and Eugeal has to call again to continue. The first thing she says is a complaint about the length of the recordings.
What really sells it is Haruka and Michiru don't move the entire time, even when the message cuts off, the phone rings again, and the message starts back up.
Funny in retrospect: Poor Sailor Iron Mouse. She looks around forlornly, wondering who could possibly have a star seed. She turns around to see a poster that announces a concert. It shows five people, namely Karajan, Michiru and the Three Lights. Guess who has a true star seed, and guess whom Sailor Iron Mouse chooses.
In the third movie, Sailor Neptune disapproves of the thought of staying a child forever, and says she can think of some "fun things" only adults can do. Sailor Uranus blushes after this.
Neptune's line in the English dub, about how a full life can contain plenty of "wonderful experiences," sounds just a little bit less suspicious.
In the manga after Chibi Usa returns to the 30th century at the end of the Black Moon arc, Mamoru and Usagi proceed to kiss but are suddenly interrupted by their future daughter falling on their heads again with a letter from Neo Queen Serenity showing us that after 1000 years, Usagi never changes.
In the anime, Usagi first meeting Chibi-chibi. She asks her her name, and only gets a "chibi" as a response. She tries asking her where she's from, same response, and where her family is same response, and, you get the idea.
In episode 159, Mamoru goes to a very fancy restaurant with Chibiusa. Usagi and the others follow. Hilarity Ensues.
When the waiter asks the girls what they want to order, they say in unison: "ice water". Only Ami timidly orders apple juice.
Then, Chibiusa gets the feeling that someone is eavesdropping on her and Mamoru. Sure enough, when the camera pans out, you can see the girls surrounding their table, hiding behind various items they've moved closer: a flower pot, a painting, a statue, and a PIANO.
The end of episode 2 where Serena is writing a love letter to Tuxedo Mask and tells Molythat it's homework. Miss Haruna arrives in the room just as she says this and chases the two of them around the room.
"You? Did homework? This is an event. Give it to me, I'll read it out loud to the class"
The Japanese is arguably funnier. Haruna knows it's a love letter, but is just as excited to read it.
In the dub when Serena finds out that she is Neo Queen Serenity. Her delivery is priceless.
"You mean I'm going to be Queen of an entire city? And my teacher says I won't even get out of grade school"
From the dub when Rini tries to open the gate to the future and instead turns the inside of a restaurant into zero-gravity. The Sailor Scouts are all inside and Raye's head floats up Mina's skirt.
Lita: Hey, we're all floating!
Raye: (head under Mina's skirt) What a totally weird feeling.
Mina: Uh, do you MIND!
Katzy's first episode where she gets into a fight with Sailor Moon.
"You'll pay for that! No one but me touches my face!"
Luna is incredibly offended when Sammy makes a comment about Rini floating ball.
Sam: That ball looks a lot like Luna.
Luna: It. Does. NOT!!!
In a nice prelude to Azumanga Daioh's shoe fortune scene, Episode 2 of the Japanese version has Usagi deciding to toss her shoe to determine where to go read her fortune. It lands right-side up...onto Mamoru's head. The look on his face is just priceless. Also that her shoe ends up on his head more than once.
Usgai/Sailor Moon: If you hadn't said anything, they wouldn't have noticed.
From Ami/Mercury's intro episode, Usagi and Luna freaking out when Mamoru gets suspicious about Luna talking. The two run a short distance and when they see Mamoru still standing behind them, they high tail it for several blocks.
The first episode as well. Supposedly, crying does help a lot...
Sailor Moon: *freaking out because of nearly getting struck to death*
Tuxedo Kamen: Crying won't do you any help, Sailor Moon!
Sailor Moon: BUT I CAN'T HELP IT!! WAAAAHHH!! *turns out the cry gets weaponized and disabled the Youma*
Youma: STOP CRYING!!
In the dubbed version Luna sums things up just perfectly as she spies on Serena after she met Darien for the first time.
Luna: I don't believe it. This crybaby she's the one!?
Episode 92. Usagi gets heartbroken when she thinks Mamoru forgot her birthday...until she realized that not only did she not tell him when her birthday was, but she didn't know when HIS birthday was!
From the dub of Chibusa/Rini's first episode. Serena finds Rini inexplicably in all of her family's photos. Finally losing it after the day she's had, she bellows to the sky "Is it the world that's going crazy or is it me!?". Her mother calmly replies "it's you, dear."
From the dubbed episode of Rei/Raye's first appearance, Serena and Amy notice all the girls who have Cherry Hill charms look really tired and out of it.
Serena: I want the same kind of charm they have!
Amy: Why Serena? No offense but you're already naturally kind of out of it.
Serena: Aw Amy I'm not out of it I'm daydreaming.
In the episode twelve, after Rei and Usagi get into another argument at the end, Ami just pulls a book out of nowhere and begins reading, which is funnier considering they're on a wrecked ship in the middle of nowhere.
When the girls go to the party with the English exchange students (French in the dubbed version) the girls all want to dance with Haruka. Rei, Minako and Makoto start playing rock paper scissors to see who goes first. Usagi gets a dance and the rest of the girls and guys at the party are crowded around Haruka. In the background we can see that they are still playing rock paper scissors and now Ami has joined in.
In the third movie, the scouts are hypnotized by the enemy and hallucinate they are in a gingerbread house. Serena eagerly grabs something to eat, only for Tuxedo Mask to shatter the illusion and reveal that she is about to eat a hanging plastic ring from a playground. She laughs, and pushes it away, only for it to smack her back in the face.
Ami gives a backhanded, and hilarious, compliment to Usagi when Minako insults her cooking. Minako hopes Mamoru will be okay after eating Usagi's cooking, and Ami comments that Usagi cooked her food for so long, there was no way harmful bacteria could be on them.
In episode 8 when Sailor Moon and Sailor Mars were caught in a hallucination, and were dreaming that they were riding a merry go round with Tuxedo Mask. Cue the scene going back into reality and you'll see Sailor Moon and Sailor Mars riding on brooms instead!
Dr. Tomoe of the anime was by far the most entertaining Big Bad of Sailor Moon, from his evil laughs to running on treadmills while on the phone.
In one episode, he is hamming it up to Eugeal/Eudial on the phone and squeezes the receiver so hard he cracks it. Eugeal then calmly says, "By the way, it sounds as if you've damaged the phone - I'm afraid we'll have to charge you for that."
The Running Gag wherein Eugeal's car would appear out of increasingly absurd places. One of the more humorous ones? When she crashed a party to steal her target's pure heart... and called the car through the wall, before driving away from the property damage.
In the manga, after Chibi-Usa and Usagi get their ages swapped, the other senshi arrive at the hospital where Mamoru is. Upon seeing the aged Chibi-Usa, they all instantly accuse her of being Black Lady.
I don't know if it's a result of the Italian dub or not, but in episode 33 one of the voices cheering on the fake Sailor Moon is Minako's. Then Minako is seen watching a whining Usagi, her face saying: "If that is Sailor Moon, ''who the hell did I just cheer?".
Later in the episode, Usagi declares that nobody can use the name of Sailor Moon without her permission, and wants to find the fake and make her pay the royalties.
Gets funnier on a meta level: this is Minako's debut episode, and in its manga counterpart did just that, sort of: she told everyone she was the Moon Princess.
And in the S season, Minako not only shows up disguised as Sailor Moon without permission (and without paying royalties), but she used Usagi's Transformation Pen for it, and declared that someone like Usagi could never be Sailor Moon to her face.
In episode 106 we see Eudial jumping on her car to attack the victim of the week. Then we have a cut to a zoo, with a monkey in a cage, an hippopotamus in a pond, an elephant that is walking in a building... Then stops, steps back in a hurry and barely avoids being ran over by Eudial's car.
Eudial's car popping out from the most improbable places is a Running Gag.
Mamoru (disguised as Tuxedo Mask) gets a minor one in English Episode 23 when he gets a silver crystal. Zoycite says "Hey, that's mine" and Mamoru responds by saying "Sorry, I don't see your name on it anywhere, loser!" then runs away.
In episode 13, after Mamoru insults Motoki, Usagi cries loud enough to cause a scene and embarrasses Mamoru. He gets criticized by some people for making a girl cry and quickly runs away.
Episode 51, Sailor Moon R. Dogged Nice Guy Umino saved the Senshi, Naru and the others a spot for the sakura-viewing festival by staying since last night. Everyone's shocked. Ami asks, "Weren't you cold?", but he came prepared with a sleeping bag, and demonstrates by crawling in and zipping up to show how cosy he was. Cheerful music starts playing. Besides, it's for everyone and Naru. Naru, seemingly touched, reaches out towards Umino... there's a beat... the music stops and she zips the bag up over his face, then turns back and invites everyone to sit down to lunch.
In one episode of Sailor Moon S, a Remless/Lemures named TogeToge Jo is summoned...and while it's arguing with CereCere and is forced to eat a dream mirror (She thinks it's icky), Sailor Moon and Sailor Chibiusa avert the Talking Is a Free Action and attack her. The dub actually makes this quite humorous with her last words.
"I haven't even done anything!"
Episode 31 has a cat version of Tuxedo Mask who saves Luna. Instead of a rose, he throws a fishbone, and they even play Tuxedo Mask's theme when he shows up.
All of the Inners reveal themselves to be members of the Three Lights Fan Club (with Ami of all people having the lowest number of the group). What do they do when they meet the #1 fan? They bow down to her!
A lot of Usagi's romantic fantasies in the first season and first arc of R fall under this. One of the most notable ones is when she imagines she's flying hand-in-hand with both Tuxedo Mask and the Moonlight Knight when, in reality, Mamoru and Shingo are just trying to get her to keep moving inside a virtual reality game.
The girls finding out that Haruka is, indeed, a woman. Made doubly funny after the fact that Usagi and Minako had been stalking her the entire episode.
The episode where Makoto loses her confidence and goes to train at a mountain temple has a few:
Finding out that Usagi snuck out of the house to ditch Chibi-Usa and leave her behind. Chibi-Usa throws her Luna-P ball down in frustration and the scene transition shows Usagi getting hit in the face.
A brief one that illustrates the girls' personalities perfectly at the same time: After a pillow fight, they all fall asleep in the same room and, while Makoto slips out, the camera shows a shot of the rest of them sleeping. Rei has half-fallen out of her futon, Usagi and Minako are holding hands, and Ami is just peacefully resting without having moved at all during the night.
One by one, the girls (including Ami) revealing that they, like Usagi, had brought their swimsuits along in order to go hang out at the resort hotel near the temple rather than train (except Makoto who was already out training). This is even after Luna and Artemis had been talking about how Ami was the only diligent one of the group.
Makoto attempting to meditate with the monk Kakusui under a freezing waterfall. She asks him for advice and he remains quiet for several seconds. When she begins to wonder if she really isn't cut out for the rigor, he sneezes and reveals that he was so cold and numb that he couldn't even talk.
A Lemures in SuperS traps Sailor Moon in a magic tutu that forces her to endlessly perform ballet moves. She ends up breaking out of it because it's too small for her.
In the short movie, Ami's First Love when she receives a love letter in school.