It's a little moment, but in the song "Chu-Chi Face" when Baron Bomburst and the suit of armor snap their fingers in frustration at the same time. Of course the suit of armor isn't animate, but the laws of reality can be broken for a good gag, right?
I always loved the spies trying to be English with horribly out-of-date information as to how the locals really dressed and acted.
Not to mention their classically slapstick antics while trying to capture Caractacus and Chitty, including mistaking Grandpa Potts for the inventor — hilarious.
Truly repeatedly hitting Caracutus during their duet. Most of the times seem possibly like accidents, but the last one clearly happens on purpose.
The "Posh!" musical number; Grampa has just been abducted, little house and all, by Baron Bomburst's zeppelin, and at one point the balloon starts to drop, forcing the crew to toss various things in an attempt to lose weight.
Baron: We are still losing height!? But zhere must be zomething we can throw out!
Agent: But there's nothing we can throw out!
Other agent: Yes, there's nothing... (Chuckles nervously when he sees that he and his fellow agent are being stared at)
After they're ejected:
Other agent: What do we do now?!
Agent: Start swimming!
Other agent: I don't swim!
Agent: THEN START DROWNING!
And then they show up in Vulgaria at the end, see the fighting, and just start swimming back out to sea. Apparently, he learned.