In Episode 2, after most of Team C pokes fun at the StuG 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Bonus funny points for the historical reference being intentionally wrong.
Also from episode 2, the introduction of the instructor.
That's the principal's car!
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!
Episode 4/The fourth OVA.
The Anglerfish Dance complete with costumes and a moving platform for maximum exposure.
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.
Episode 5. 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.
Episode 6. Alice'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 Alice. 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 when she and Nonna begins singing Katyusha.
Miho's idea to improve morale when the team is trapped in a church by Pravda's team? The incredibly embarrassing and demoralizing Anglerfish Dance. It works.
The Dance is broadcast on a HUGE screen for all to see... including Maho, Shiho and Yuri. 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 "STOP!" is equally funny.
Kay:(waving to Yukari) Hey! "Sergeant Third Class Odd Ball"! Yukari: She's onto me!
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: Hit me if you must. (Saori kicks her with extreme prejudice) Hana: Could you be a bit gentler?
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.
In Episode 5, 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.
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?
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.
The Hurricane of Puns shown when the School Council go to a restaurant. Every single line the store owner says has the word "Win" mangled into it.
When they attempted a third trolling session Maho caught up with their hijinks and ordered one of her tanks to open fire. Fortunately they survived that.
Kuromorimine's Panzer III is trailing behind the Maus during the finals, 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.
Maus commander: Hey! You! Light tank! Get out of there! Noriko and Akebi: Don't wanna! Also, the Type 89 isn't a light tank... (beat) it's a medium tank.
Rabbit Team's grand strategy against Kuromorimine's Elefant: Go round and round the same block in a smaller, faster tank until you catch up to them from behind.
Sodoko deletes all records of Mako coming in late, who proceeds to break character, to put it lightly.
Real Life example: Construction signs in the real life Ōarai were spotted with this statement: "Construction due to damage from Senshado match. Please contact Senshado 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)