The First Inspector: You two have been with me through some terrible times indeed. When the Blorgons kidnapped me in the 27th century, you were by my side. When the Spanish Inquisition interrogated me in their torture chambers, you were there. When the Radarmen almost marooned me off the Scutum-Crux galactic arm, you were there, too. When somebody left the time booth door open during takeoff, there you two were still. [Beat] Do you know what? Bart and Irma: [Together] What, Inspector? The First Inspector: I think you're bad luck!
BTV's production department's attempt at depicting an Ejector Seat in action during "The Retirement Home of the Circuit-Chaps" led to one of the show's more hilarious/infamous Special Effect Failures.
From "The De-Faced Doppelgängers":
The Inspector: Why are you wearing sunglasses at night, my good man? Faceless Goon: Why don't you mind your own business, mate? The Inspector: My own business would bore me to death. I mind other people's out of self-presevation.
The Inspector at the end of "The Cube in Time" (and his second season), when Aiden urges him to "celebrate" their hard-fought victory. Nowhere else in the universe will you see Christopher Lee playing an ocarina and doing, just for a moment, a little soft-shoe dance in Wellingtons.
The bit with the pot of blasting powder and the torch in "The Revolution". Classic enough that Lee recreated the moment decades later when playing Saruman in The Lord of the Rings.
From "The Spores of Doom":
Aiden: Do you think you can beat the Sulphur Squadron Leader in a duel? The Second Inspector: I can best him blindfolded. Aiden: What if he's not blindfolded? The Second Inspector: Then it might be a bit more difficult.
From "The Phobia Parasite" (the second appearance of the Sergeant):
The Third Inspector: Can you describe the person you saw last night? Witness: He was male Caucasian with a moustache. About six-foot-four. The Third Inspector: It couldn't have been the Sergeant—his moustache isn't that big.
"The Kittens" very possibly had the most unintentionally funny "villains" of the entire series, but the Inspector's various gags with that bit of string are genuinely hilarious. Marvel at Bernard Fox making something out of almost literally nothing.
The Inspector to the Sergeant as he finally arrests him in "The Space Creature":
The handcuffs are tight because they're new. They'll stretch out after you wear them a while.
The Third Inspector and his long-time Associate Yosif had a Running Gag about his mental abilities:
In "The Talons of Asox":
The Third Inspector: Not to worry, I have a photographic memory. My mind is like a camera. Yosif: Always going out of focus?
In "The Ocean Demons":
The Third Inspector: Not to worry, I have a photographic memory. My mind is like a camera. Yosif: It needs developing?
In "The Blue Ruination", during their Tear Jerker parting scene:
The Third Inspector: Not to worry, Yosif, I'll never forget you. I have a photographic memory. My mind is like a camera. Yosif: [Trying Not to Cry] Can you still find film for it?
The clashes between Fourth Inspector and the Superintendent were always good for a laugh during the programme's Exo-Pol years.
The Inspector to the Superintendent after the Final Battle of "Exodus of the Blorgons":
Before the Blorgons come back with their next criminal enterprise, would you kindly set your watch five minutes fast so you could arrive slightly ahead of the nick of time?
When the Inspector is trying to take into custody a computer terminal that he suspects has been suborned by the Digifleet in "Reappearance of the Circuit-Chaps":
The Superintendent: On what grounds are you going to arrest that machine? The Inspector: Probably cause. The Superintendent: Probably cause? What's that? Don't you mean probable cause? The Inspector: No, I do mean probably cause—probably cause it's winding me up!
The opening to the "Horror of the Asterozoids" serial apparently refers to an incident in between Seasons 12 and 13:
The Superintendent: I won't stand for any more antics like yours last summer in Westminster. Is that clear? The Fourth Inspector: When I see the last Pharaoh of Ancient Egypt appear in the middle of Regent's Park in full view of 1,200 people, I must investigate the chronal anomaly. Am I clear? The Superintendent: That was the Open Air Theatre's production of Shakespeare'sAntony And Cleopatra!
The Fourth Inspector's Witty Banter with Mary Sue Brown was one reason she always tops the lists of fan-favourite Associates:
Andrew Royd a.k.a. Android Z1: Brain Trust, Ltd. has ensured that all the people in our Human Resources Department operate strictly within the law. That's the truth, pure and simple. The Fourth Inspector:Ahem. I'm not simple, and she's not pure. Mary Sue: Besides, Inspector, the truth is rarely pure and never simple.
From: "The Robot Revolution":
DI Gary Mulligan: [As robot versions of the Inspector and Mary Sue emerge from the aliens' fabber] You make a lovely couple. The Inspector and Mary Sue: [Together] A lovely couple of what?
From "The Moribund Mind":
The Fourth Inspector: This proves that the midget who escaped from Wormwood Scrubs yesterday is really a clairvoyant alien! Do you realize what that means? Mary Sue: There's small medium at large?
Lots of glorious black humor in Terror at Tooth Point, starting with that title; while the action does in fact take place on Tooth Point, it turns out the villain of the piece is a Classical Movie Vampire.
One specific example: when Reena joins Morbius's collection of hypnotized thralls, instead of wearing a long flowing gown like the rest, she switches to a fluffy white version of her usual Jungle Princess bikini.
Veneziana, Nymeria and Thorough's reactions when the newly metamorphosed Inspector cheerfully emerges wearing her... rather striking... new outfit, topped with that hat, particularly Veneziana's terrified little squeak. The Inspector then applies some lipstick with suspicious confidence and expertise...
Inspector: There! How do I look? Thorough: ...Breathtaking?
Interplanetary Rescue Patrol: Come in, unidentified spacecraft. What is your position? Repeat, what is your position?
The Inspector: My position? Well, I'm a Detective Inspector of the Infinity Knights from the planet Kayaclasch in the Boötes Galaxy."
This exchange, in reference to one of "The Stinging Nettles":
Thorough: That thing looks incredibly dangerous. Inspector: Indeed! Nymeria: ...What are we going to do? Inspector: Poke it with a stick!
The Inspector-Muppet in "Mindscrew". In addition to being adorable.
The Fifth Inspector's blink-and-you'll-miss-it cameo in "The Only Inspector":
Door gets yanked open
"Oh! Hello there!"
"No, no! Too soon! Too soon!"
Door gets slammed shut
Pretty much the entirety of "Spectre Night". Sure, a lot of fans think it's an overly simplistic Neo-Creationist parable, stretched over far too many episodes, but the writers more than make up for the thin plot with the sheer volume of jokes for the Inspector, e.g.:
As my old playwright friend Bernard Shaw once asked, if the world were only one of God's jokes, would you work any less to make it a good joke instead of a bad one?
Evidence is for people who find reality more interesting than anything they could make up themselves.
Biology is the only science in which multiplication means the same thing as division.
Under these rigorously controlled conditions of temperature, pressure, humidity, etc., etc., the organisms, I predict, will do exactly as they please.
[To the Intelligent Designer aliens] You probably also think that homology is the study of real estate, serology is the study of knights, bacteria is the rear of the cafeteria, and the hippocampus is where hippos go to university.
If Darwin was right after all, you should probably figure it out in a few million years.
"I say, you certainly don't look like a zombie, poppet!"
From "The Creation of Earth":
The Inspector: Orbitting that star over there is the Dodo Planet. There's a strange and terrible story about how it got its name. Lily: What is it? The Inspector: Nobody knows. We know only that there's a strange and terrible story.
The Ninth Inspector is much more Sarcasm-Blind than either the Fifth or Eighth Inspectors, e.g. this exchange in "The Humans of Westminster":
The Inspector: Detective Inspector, Chronospatial Inquiry Division. I must get inside. Buckingham Palace Guard: Of course you are, and I'm the Queen's uncle. The Inspector: Your Grace, I must get inside!
The wonderful scene in "A Spacetime Musical" where the Inspector replaced James Haggard's nuclear missiles with a grove of watermelons. The lyrics and choreography to that number were a hoot, as well.
The Inspector goes off the rails momentarily in the middle of his Patrick Stewart Speech in "The Xmas Rebellion":
The Inspector: I ask you, is that fair? Of course it isn't—fair is a place where you jump around in the bouncy castle, get sick on candy floss, and step in monkey poo. Or was I asking rhetorically just then? Or right now?
The Inspector has a little trouble adjusting to early 21st-century London police in "The Nicodemus Examination" after his stint with Exo-Pol in the 1970s:
The Tenth Inspector: Did Doctor Nicodemus leave a ransom note about Joanna? DCI Drake: Yes, I sent the note to our crime lab. It's demanding one million Euros. The Tenth Inspector: Why would your crime lab demand a million Euros?
Eleventh's gleeful reaction in "Cavemen on a UFO" to winning the Cavemen vs. Astronauts Debate with Angie and Geneva over whether Ancient Astronauts abducted Neanderthals to crew their spaceships when they locate one exploring Bode's Galaxy in the 35th millennium BCE: