The jaguar from the original short, both in how he's animated and the way he says "I need space!" Apparently, this was a friend of Nick Park's from Brazil who was visiting England and hated it.
One of the crows (or ravens, or whatever they are) in the Birds episode apparently thinks birds fly the way Peter Pan does: By having a happy thought! Being happy most of the time. And this is coming from a bird associated with darkness and grimness.
The Vet episode is comedy gold, but one bit that stands out to me is the horse on the table, saying that he's taken a lot of pills in his life, "About 23,000." Now that's a funnily specific number.
US example: The bulldog couple have an Odd couple thing that is so funny.
"What am I afraid of? Anyone who knows more about me then they should. And I married her, so I live in constant fear." (He says the last part with a smile and his wife right next to him.)
Another moment from the US series, the gerbil (or possibly hamster, it's hard to tell) talks about a fear he had as a child that it both hilarious and slightly horrifying.
"As a child I was afraid that my parents were going to murder me. And once I had lived long enough and they didn't I guess I had a pretty much a trend going towards not being murdered. So I got over that pretty quick."
In the US series, one of the elderly lions' answer to the question "Have you ever killed a man?" The pauses as he thinks about his answer are priceless.