Paddington tells Mr. Brown his "bear name" with a growl. Mr. Brown tries to growl similarly, leading to this:
Paddington: Mr. Brown, that is extremely rude.
Millicent's Fantastic Racism about bears and the horrors they'll bring to the neighbourhood:
Millicent: Drains choked with fur. Buns thrown at old ladies. Raucous, all-night picnics.
The security guard at the Cartographers' Guild, with his bewildering attraction to Mr Brown in a dress.
Even when Mr. Brown is fleeing out of the building, rapidly shedding his disguise, the security guard still yells "Stop that sexy woman!"
The explorer naming Aunt Lucy after his dear, departed mother, and Uncle Pastuzo after... an exotic boxer he met in a bar once.
Almost everything about the explorer in his film. From his sheer stereotypical Britishness, to his "essential" supplies including a "modest" timepiecenote actually some kind of grandfather clock and travel piano, to his flustered response when the bears start messing with his possessions, to his reaction when the bears demonstrate an ability to learn English:
Aunt Lucy:Lun... Dun... Explorer:Good. Lord. Now try "Stratford-Upon-Avon".
When Mr. Curry sees Millicent the first time. Lionel Ritchie's Hello just nails the scene.
After Mr. Curry telephones Millicent to inform her that Paddington is home on his own, he opens the door to her to reveal that he has changed into a terribly outdated tuxedo in an effort to woo her.
The sequence Paddington accidentally chases down a pickpocket, trying to return the wallet he dropped. Especially this line from the culprit;
Some kind of....tiny police bear!
Also in the same scene, various puns such as the satnav ("in 100 yards, bear left!" [Paddington is flying through the air 100 yards to the left]) and Judy's English class has the immortal Shakespeare stage direction "Exit, pursued by bear".
The flashback explaining how Mr. Brown changed from a wild, motorcycle riding bad boy to an uptight, super concerned man when he became a father. We see him drive in on his motorcycle to the hospital, the baby is born over night, and literally the next day, he's completely changed to his present-day self.
Mr Curry has a change-of-heart and tries to inform the Browns about the danger Paddington is in by calling them... anonymously:
Mr. Curry:[Unconvincingly trying to disguise his voice] Good evening. This is... an anonymous phone call... Mr. Brown:[Nonchalantly] Oh, hello Mr. Curry. Mr. Curry:[Losing the voice] It's not Mr. Curry! It's Mr... [Re-adopting the voice]Burry. I have some news concerning the bear. Mr. Brown:[Listening to the phone in horror]... What?!... What do you mean?! Mrs. Brown: Who is it, Henry? Mr. Brown: It's Mr. Curry, doing a silly voice. Mr. Curry:[Through the phone] It's Burry!
When Paddington runs away, Mrs. Brown goes to the police station to file a missing bear report:
Mrs. Brown: He's about three foot six, got a bright red hat on and a blue duffle coat... and he's a bear. Police Officer:[Taking notes and shaking his head] It's not much to go on. Mrs. Brown:[Taken aback] ...Really?
The meta-joke here is that she's just described the most famous and recognizable non-stuffed fictional bear of the 20th century.
The running gag of people not being surprised by the existence of a small talking bear in a hat.
At the climax, when Mr. Brown is trying to get Paddington's attention from a ledge outside the room where Paddington is waking up from a stupor:
Paddington: Is that you, God? Mr. Brown: What?! Paddington: It's just... you sound a lot more like Mr. Brown than I imagined.
Judy's Chinese lesson tapes recite regular sounding phrases for learning a foreign language, then one says:
Audio Tape: I have been accused of insider trading, and require legal representation.