When Archer tells you about his greatest triumph, saving 10,000 people from a nuclear reactor meltdown, the protagonist says that at least he got a good tan out of it. Archer's response?
Archer: "Ha. Ha. Ha. I don't think even Rin Tousaka could say something so cruel."
Talk to the Student Council NPC in charge of the Library on the 3rd day of the 2nd week. She tells you about using the triangle button to skip dialogue, and starts, in her own words, "whining relentlessly about my boyfriend" so you can practice. The whole rant is comedy gold.
The kicker is if you sit through the whole thing. After a Beat, she adds that you can review any conversation you've just had by hitting square, but "you shouldn't do that now".
If your Servant is Saber, the enemy Archer repeatedly insults her at one point, and she fixates on the fact he called her 'princess'. Apparently, she's worried you'll start using that as a pet name.
If you choose to save Rin, talk to Archer during the 4th week. It begins with Archer falling out of his chair after you ask your first question. All but the last question you can ask after that give some pretty hilarious responses.
During one of your early conversations with Saber, you can ask about her True Name. She actually starts to tell you, then suddenly realizes what she's doing and backpedals hilariously.
Saber: "Erase all thoughts of Lu- from your mind!"
Late in the game, one of Saber's conversations mentions that she once wrestled a lion while naked. Hilariously, the protagonist isn't bothered by the wrestling or the lion part, but it's the naked part that throws them off.
"Isn't there something obscene tucked away in the middle of that line?!"
Just like on the 4th week, if you chose to save Rin, you can talk to Archer on the 6th week. The first question you ask him is "Is Rin your ex-girlfriend?" and it gets funnier from there. For starters, the protagonist calls him a "playboy of the highest caliber".
When asked if you prefer comics or manga, you can select bande dessinee. Response: "...How very hipster of you."
The lunch scene is awkward comedy straight from a school manga; of the Yuri Genre no less, if your MC is female. Your Servant clinches it, from Archer bluntly informing you that your oblivious reaction was "a dick move" to Caster's later conversation in your room, including one hilariously fourth-wall-breaking option.
During a Week One conversation with Archer, one dialogue choice results in the MC despairing of their bitter, cynical, sarcastic new Servant and wishing you could swap him out for a "beautiful, graceful, respectful" one. Archer informs you that they were fresh out of those, but not before cracking up at your expense.
One of the NPCs in the courtyard has a Command Seal tramp stamp. Also, he's male. Yeah, he's not too happy about it.
On Rani's path, your conversation with her in the Infirmary is interrupted by Gatou, who doesn't seem to know the MC is a Celibate Hero.
Gatou: I did not wish to believe it, but a clandestine tryst is being held!
Gatou: Young man! Follow the path of abstinence and righteousness!
During Rani's mana transfer, the typically-unshakable Archer is... somewhat shaken. Rani, on the other hand, is indifferent until she realizes you're peeking.
Archer: Wait a second, I thought you were from Egypt, not Babylon! You don't have to touch me there...
An early conversation with Archer leads to the following narration from the player character:
Archer may be a pain in the ass, but all he's doing is making me think about what being a Master truly means. He's not "topping from the bottom."
Archer: "*Sigh*... Suddenly, I don't feel like doing it."
A scene with Lancer after the battle with the Shinji Tank, which explodes soon after. When Lancer was about to be caught in the explosion, the protagonist saves her. The result, Lancer becoming disoriented, revealing her deredere side, complete with flowery background. Everyone who was watching (Rin, Rani, Sakura, and whatever Servant you selected even Gilgamesh!) was not amused... except Archer.
Lancer: "Hey, darling. I had a weird dream. You and I, we were fighting, and after some evil, heartless women brainwashed you, I was about to die in the blast of an idiot's malfunctioning weapon. But then you came running, saved me, and told me 'If you were to die, I would not want to keep living either, baby."