You got to admit that the scene at the train in which Vlad is counting Anya vs Dimitri and Anya is winning by a LOT has to give you at least a chuckle. If one were to count the score—"Dimitri: 3 vs. Anya: 30."
From around the same part of the film:
Anya: Do you really think I could be royalty?
Dimitri: You know I do.
Anya: Then stop bossing me around!
When Dimitri tries to wake up Anya on the train.
Dimitri: I think you broke my nose!
Anya: I'm so sorry...Oh, it's you. Well, that's okay then.
During their efforts to escape from the out-of-control train:
Dimitri: Give me a wrench, an ax, something!
Anya puts a stick of dynamite into his hand.
Dimitri:[doubletakes] That'll work.
"What do they teach you in those orphanages?"
Anya's new blue dress and her reaction to it:
Dimitri: I bought you a dress.
Anya: You bought me a...tent.
Dimitri: What are you looking for?
Anya: [with her head inside the dress] The Russian Circus! I think it's still in here!
Rasputin's freak outs when both his plans to kill Anya fails.
(Anastasia has left the train cabin after an argument with Dimitri.) Vlad:(gleefully to Pooka, Anastasia's dog) Oh no! An unspoken attraction? Dimitri: ATTRACTION?? To that skinny little brat? Have you lost your mind?
And the end, where Anya accidentally smacks an injured Dimitri in the face and then glomps him out of the happiness of seeing that he's still (also a heartwarming moment).
Before they meet Anya, Dimitri and Vlad are holding an audition for girls to play Anastasia. One prospect is a middle-aged woman with a fur coat and cigarette who says in a sultry voice, "Granmama! It's me, Ana-STASIA!" Complete with a burlesque hip-bump and a rimshot. Their reaction is a combination Head Desk and Face Palm.