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In Episode 2, after most of Team C pokes fun at the StuG III they've been assigned to, Erwin (who more or less knows of her soul name's exploits in both World Wars) lists its accomplishments, and then, while pointing off into the horizon, delivers this line:
Apologize to the Finnish!
Bonus funny points for the historical reference being intentionally wrong.
Well, only by one war. StuG III didn't serve in Winter War. However, they saw use in Continuation War.
Also from episode 2, the introduction of the instructor.
Isn't that the headmasters car?
''Now they've gone and done it."
Aya asks the internet how to start the tank, and for her trouble receives responses such as "Google it, noob!" and "You start by taking off your clothes."
In Episode 3, Mako's reaction to learning that they need to be at a friendly match by 6AM... which means she needs to get up at 5AM. She nearly facefaults, before trying to quit The Team. To make sure she wakes up on time, said team drives the Panzer IV to her house and fires a blank shell from the main gun... which just barely wakes her up!
Since early tanks didn't have internal radios, it's impossible for the commander to verbally communicate with the crew over the tank's engine noise, so she has to give commands to the driver by kicking her to indicate which way to turn. Hilarity Ensues (not to mention, considerable SadoMasochism-flavored Accidental Innuendo) as Saori tries this on Hana in episode 2.
Hana: Kick me as hard as you can! Saori: What?! Fine! Then LEFT! (Saori kicks her with extreme prejudice)
Hana: Could you kick me a little softer?
In the manga, they mention that was done mainly with the smallish Japanese tanks. Cue Saori's legs not reaching Hana's shoulders in their roomier German tank and Miho handing her a stick to poke Hanna's shoulders with.
There's a frantic repeat of this in Episode 11 by the Kuromorimine crews, as the Student Council goes Hetzer trolling.
Episode 4/The fourth OVA.
The Anglerfish Dance complete with costumes and a moving platform for maximum exposure.
Not to mention when Anzu has the student council participating in the humiliating dance, unlike everybody else, she is seen looking VERY happy and enjoying this, despite that she herself is part of the humiliation.
The owner of an Ōarai (the city, not the school) shop that gets smashed into by a Matilda positively Squees at the opportunity to remodel it with the insurance money, prompting a fellow shop owner sitting next to him to pray that his shop also gets wrecked!
Shop owner 1:"My shop!" (beat) Now I can have it renovated!" Shop owner 2:"It's a good omen." Shop owner 3:"Fingers crossed they'll also crash into mine!"
Bonus funny points for being Truth in Television; the real-world counterpart of that shop received a HUGE number of customers because of it.
Mako snarkily responds to Erika's description of an unwritten rule that schools with no chance of winning shouldn't enter the tournament by saying that it would be utterly humiliating if one of those schools defeated a well-funded one like Kuromorimine.
Erika: There's an unspoken rule that everyone knows: schools that will just embarrass Tankery stay out of the tournament. Mako: It'd sure be pretty embarrassing if one of those teams beat the veterans that all sat around in their ivory towers coming up with that rule.
The Panzer Cafe. Everything from the bell to call the waiter over being a tank that makes a firing noise when pressed, to the military-style costumes they're wearing to the cakes ordered in the shape of tanks. They're brought on the tables by automated scale models of tank transporter trucks!
All being various bits of Truth in Television, though perhaps not all in the same restaurant. Themed restaurants with waitstaff in uniforms (maids, ninja, EFSF or Zeon soldiers, etc.), themed menu items, and some sushi restaurants have little automated trains that deliver orders to your table.
adding credits to her infiltration video.
The part where she's about to change, realizes that the camera's still on, and shuts it off,
Also the answer she gives when asked for identification.
Kei:(waving to Yukari) Hey! "Sergeant Third Class Odd Ball"! Yukari: She's found me!
Episode 6. Arisa's hilarious breakdown as the Ōarai tanks chase her lone Sherman. Then the Ōarai tanks end up being chased by the rest (four more) of Saunders' Shermans, while they're chasing Arisa. All while the History Team keeps quoting battles where tables were turned and/or the siegers became the besieged.
Episode 8. Katyusha's childish antics in general, especially in the original Japanese version, when she and Nonna begins singing Katyusha.
Also when she and Darjeeling start the episode having tea, after St Gloriana's has been knocked out by Kuoromorimine in the semi-finals Darjeeling warns them to practice for the semi-finals, whilst Katyusha says that Oorai are just a no-name school. Darjeeling points out that they are led by a Nishizumi:
Katyusha: Wha-?! You didn't tell me that, Nonna!
Nonna: I've told you several times.
Katyusha: Well this is the first I've heard of it!
Katyusha: Surely you didn't come all this way just to tell me that.
Darjeeling: Of course not. I just wanted to have some of this delicious tea.
Miho's idea to improve morale when the team is trapped in a church by Pravda's team? The incredibly embarrassing and demoralizing Anglerfish (Ankou) Dance. It works.
Miho: Everyone, sing with me! I'll do the Ankou Dance!
Momo: Hey, that's going to have the opposite effect!
The dance is broadcast on a HUGE screen for all to see... including Maho, Shiho and Yuri (Hana's mom). The expressions on the latter two's faces are priceless.
The surprise on the Pravda messenger's face when she walks in the middle of the dance number to ask for Ōarai's answer to their ultimatum and is ignored until she screams "HEY!" is equally funny.
Kuromorimine's Panzer III is trailing behind the Maus manically boasting about the latter's impenetrable armor. Said Panzer III promply gets taken out by a stray shot when it swerves away from its companion's protection.
OVA 1's swimsuit-choosing scene is an extended funny moment. From the point they enter until they exit the store there is a hurricane of rapid-fire jokes, from Yukari's swimsuit of choice being military diving equipment, to the themed suits chosen by the other teams, ranging from loincloths to dissolving bikinis to a full-body team mascot costume. Finally topped by a heavy storm starting as soon as they get to the beach.
In Episode 1 Miho had to go to the infirmary when suffering from a Heroic BSOD and Saori and Hana immediately made up excuses to join her. In Chapter 7 of Little Army, Emi forgoes such subtlety and instead drags Miho out by the wrist in an attempt to get out of class before the teacher sees them, and when she runs into the teacher, hastily makes up an excuse that she's taking Miho to the infirmary, and Hitomi and Chihiro point out that they're following because they're worried, leaving the teacher taken aback with Blank White Eyes.
At the end of the first chapter of the Little Army manga, when the girls worry about how their parents will react to them being late due to unexpected complications on their test run with a Panzer IV, and consider making excuses.
Chihiro: Hii-chan, just say you were at my house and I'll say that I was at your house. Emi: And I was supposed to be back this late, so I'll be fine. It seems like the only one who's gonna get in trouble is Nishizumi-san. Miho: Whaaat!? G-guys, help me! My parents are so strict! Emi: You brought this on yourself. Chihiro: Miho-chan, you can just tell them you were at my house, too! Miho: ...with a tank?
In Chapter 3 of Motto Love Love Sakusen, Sodoko's reaction to seeing that her school swimsuit does not have her last name, but her much-hated nickname is priceless,
Chapter 8 of Motto Love Love Sakusen shows Kuromorimine getting invited to the Nishizumi house by Maho to celebrate Christmas. Hilarity Ensues because of their overly serious personalities, especially because they initially think that Maho called them in to answer for their mistakes during the Tournament Finals.
Jagdpanther-chan arriving late... again. She gets berated by the Maus commander for the same.
Maho proposes doing karaoke, but leaves the song choices up to the commanders since she doesn't know what would be a good song. All five of them put in their school's anthem.
Maho tries to cut a cake, but completely fails at cutting in a straight line.
Most of the commanders get models of the tanks that caused them the most trouble during the final battle; "Jagdpanther-chan" and the commander of one of the Panthers each gets a Hetzer, The commander of the Maus gets a Type 89, the Panzer III's commander gets a Maus, reminding her of her failure to protect it, bringing back bad memories (although Koume is quite happy to get a model of the tank Miho commands). Erika gets an Anglerfish dance costume, and Maho gets a large stuffed creature she describes as "cute".
Becomes funnier when one realizes that it was all Anzu's idea, and the gifts were (presumably) different from the ones that Miho intended, as evident from her reaction.
Chapter 15 of Motto Love Love Sakusen: Saori and Rabbit Team are shocked to find out that Saki seemed to have fallen in love, and do everything they can to help her, even various training exercises. They then watch as Saori meets with her boyfriend, greets her, and the person pulls down her hood, revealing herself as Duck team's Shinobu, who'd been hoping to hear Saki's voice.
In what doubles as a Shout Out, Ribbon Warrior shows us that Saunders used to have an M22 "Locust", but when Arisa wants to use it she learns they had lend-leased it to someone else. The American-designed Locust only saw service in British usage.
Real Life example: Construction signs in the real life Ōarai were spotted with this statement: "Construction due to damage from Sensha-do match. Please contact Sensha-do Association with regard to insurance and damage claims."
Another Real Life example: the following commentnote (quoted verbatim, including the line break) seen at the end of a short review of the series' ending in 4chan's /a/:
Peace and Panzer Vor!note Remember that this literally means "Peace and let the tanks advance!" (I never thought I would utter that line in total seriousness)