Interrogation by Vandalism

A character needs to get information from someone, but Cold-Blooded Torture or the Jack Bauer Interrogation Technique would make him or her seem too nasty, or would simply be ineffective. So instead they threaten to destroy...the Priceless Ming Vase!

Usually only used by heroes on characters that show a distinct regard for possessions over human life, so it seems a bit like they deserve it. However, when push comes to shove, villains are vastly more likely to actually go through with the threat.

Often Played for Laughs (see Gene Hunt Interrogation Technique), with a common punchline being the victim's retort that "that wasn't even mine". However, nobody actually does this in real life except for maybe your parents throwing something of yours in the trash. The police especially do not do this - it's illegal to destroy property.

Examples of Interrogation by Vandalism include:

Anime and Manga

Ah! My Goddess: in episode 9 of the 2005 series, Sayoko tortures Keiichi by smashing toy cars in front of him with a small hammer. She notes that this technique seems to be particularly effective against him. Note that the toy cars are hers to begin with, and it's implied that she bought them just for this purpose.
In Soul Eater, Gopher tortures Kid by vandalizing his body. He plays on Kid's Super OCD by doing things like drawing an asymmetrical pattern on his stomach in permanent marker and scratching one of Kid's body parts and not the other. He only tried this because he noticed Kid's OCD and that the usual "beat him til he talks" technique wasn't working.

Comic Books

V for Vendetta - the original comic - has V take Lewis Prothero, formerly a death camp commander, hostage and threatens him with the incineration of his collection of priceless dolls unless he tells him everything that happened at Larkhill. And then he does it anyway, driving Prothero insane and depriving Norsefire of its major voice of propaganda.

Film

Toy Story 3 (2010). After Ken refuses to tell her how to turn Buzz Lightyear back to normal Barbie goes to town on his accessories and wardrobe. He manages to hold out until Barbie threatens his precious Nehru jacket.
Blue Ice (1992): After a humiliating drug interrogation by a member of MI-5, Michael Caine pays a visit to the interrogator at his house and gets him to talk by smashing his irreplaceable jazz recordings—the victim is more concerned about losing the music than the financial damage he's taking.
48 Hrs. (1982). Reggie starts to trash the redneck bar until he gets the information he wants.
Fanboys - when caught breaking into Skywalker Ranch, they threaten to destroy an original Millennium Falcon model when cornered by Ray Park. Taken to extremes when one of the guards then threatens to destroy the original Yoda puppet in retaliation. It fails when one of the fans tries to threaten an Ewok head and nobody cares.
In Bad Boys 2, Marcus and Mike get the location of the Haitian gang's hideout by doing this to Icepick's shop.
This is the reason The Dude's rug gets pissed on at the beginning of The Big Lebowski.

Literature

Brotherhood of the Rose, a novel by David Morrell: The heroes force the English member of the Abelard conspiracy to talk by shooting his priceless roses.
In Wolves Of The Calla, Enciro Balzar pressures Calvin Tower by threatening to burn Tower's most valuable books.
Devil in a Blue Dress (1990): Easy Rawlins bashes the beloved marble countertop owned by his barman friend with a hammer to force him to tell the truth about the job he's been given. Also shows up in The Film of the Book.
Taken Up to Eleven in a hilarious moment in The Lost Conspiracy." They get in a massive hostage stand off . . .

Character A: I smash this fish!
Character B: You do that, and I'll burn this valuable scroll!
Character C: You do that, and I'll cut her throat.
Beat
Character A: Okay, this is just getting ridiculous . . .

In Meta Game by Sam Landstrom, D_Light interrogates an Analyst by smashing some of his computer equipment. The Analyst estimates that the destruction of just one monitor will reduce his efficiency by 0.5%, which is a huge blow for someone created to be obsessed with work.
A Piece of Resistance, a novel by Clive Egleton set in a Soviet-occupied Britain. The protagonist breaks a landlady's Dresden china heirlooms to get her to reveal where she's hiding the people he's after.
In Feet of Clay, Detritus uses this to coerce a troll drug smuggler (and pottery merchant) into assisting the Watch.
- Though that was more a case of "accidentally" smashing a rather valueless statue, thereby revealing a massive cache of drugs, and using that evidence of trafficking to blackmail said smuggler into cooperating.

Live-Action TV

In The Goodies episode "Scoutrageous", the two renegade Scouts whittle Tim's staves until he relents (they also damage his hat).
- Oh, it's not just that. They also threaten to take a Brillo pad (steel wool) to Tim's shiny shoes. They threatened the shiny shoes.
Used in Malcolm in the Middle, where Lois tries to interrogate the boys to find out who lit her dress on fire and put it in the toilet. One of her methods is to go through the room and begin throwing things into the trash, and another threat was to smash the television with the hammer. Does she ever get an answer? No, because it was actually Hal who did that!
Used in Neverwhere. The Marquis de Carabas uses an antique vase to guarantee his safety when negotiating with the art-loving Psycho for Hire Mr Croup. The interesting bit? Mr. Croup was genuinely concerned by the threat... He wanted to destroy that beauty, and not let anyone else do it.
Angel: Gunn once tortured a guy into handing over a priceless scroll...by juggling a set of priceless orbs.
- In Buffy the Vampire Slayer, Spike got the geek trio to work for him by holding their Boba Fett action figure hostage.
In Las Vegas, Ed Deline and Jack Keller do this to get info from an effeminate artist by destroying his paintings, virtually giving the guy a heart attack in the process.
In the Australian series Mission Top Secret, Big Bad Neville Savage uses this to interrogate a museum curator. He was even using fakes, which made stark contrast with his decision to execute the curator and his son (along with the heroes' "mentor") later in the same arc.
Parodied on Peep Show:

Super Hans: Crunchy Nut cornflakes? Very fancy. Be a shame if someone were to... spill em. Whoops!

Pie in the Sky. In "A Matter of Taste", Margaret Crabbe is abducted and locked in a cellar by the villain of the week, who thinks she knows the location of a collection of vintage wines he smuggled into the country. When it looks like things are going to become violent, she grabs a wine bottle to use as a club, and the villain calls his mooks off—because he's worried about the bottle getting damaged. Having found his weak point, Margaret proceeds to take merciless advantage.
Downplayed on Law and Order: Criminal Intent. Vincent D'Onfrio's character is always walking around grabbing things in the suspect's home, office, etc. Sometimes it leads to Eureka Moments, but usually it just annoys the person.
Blake's 7, "Bounty": Blake starts destroying Sarkoff's antique collection to make Sarkoff reassume leadership of the planet Lindo. (Sarkoff assumed Blake was a Federation assassin, and was resigned to his death, but not the loss of his antiques.)
Boston Legal: A man confessed to a priest after kidnapping a small child. Brad Chase tries to get the priest to betray confidentiality by threatening to damage a priceless wood door with an axe. Then the priest tries to stop him by holding his hand in the way of the swing...
In Burn Notice, Michael once did this while pretending to be a criminal.
Law and Order Special Victims Unit: In the episode "Dolls," the detectives are tracking a mentally unstable man who kidnaps young girls, dresses them up as china dolls, and keeps them locked up until they die of starvation. He's also something of a doll collector, so when they finally track him down, Munch starts smashing his dolls to convince him to reveal where his current captive is.
Law and Order UK: In the episode "Paradise", the detectives and CP Alesha Phillips have arrived at a store that is believed to be a front for a local mobster's operation. When the shop owner feigns ignorance, the detectives begin "accidentally" knocking over the equipment, all the while offering fake apologies, until the guy gets fed up and tells them what they want to know.
On Mystery Science Theater 3000, Crow is inspired by the episode's movie to torture Mike by pouring beer all over his most precious possession. Since Mike's most precious possession is an antique beer stein, he's not too upset. He then takes it further by noting that Crow himself is a precious possession, and Crow immediately demands that beer be poured all over him.
In one episode of Rescue Me, Tommy begins dropping Sheila's expensive house decorations on the floor until she tells him what happened when he was unconscious. She still doesn't tell him, but she makes up a pretty believable lie.

Video Games

An option in The Godfather: The Game, where the player can extort shop owners by smashing up their stores.
In Mass Effect 2, Kasumi's loyalty mission has a combination of this and Shut UP, Hannibal, where Shepard can interrupt the villain's ranting by casually shooting one of his priceless statues.

Web Comics

From The Adventures of Dr. McNinja:

Gordito: Oops. I just accidentally poured out AAAAALL of your dish detergent.
Dr. McNinja: I don't think he did it accidentally at all!

In xkcd, our hero gets out of visiting his librarian girlfriend's family by slooowly destroying the spine of a hardback book (that he bought himself).

Western Animation

In the Powerpuff Girls episode "Collect Her", Lenny Baxter is a collector of Powerpuff Girls merchandise who eventually captures the girls themselves. To get him to tell where he has them stored, the Professor and some kids start tearing open all the collectibles he has, ruining the collectors' value.
In one SpongeBob SquarePants episode, Spongebob and Squidward think that Mr. Krabs is a robot. They destroy Mr. Krabs' stuff, because they think that they are his robot friends and that would make him confess that he's a robot.
Happened in Freakazoid!, where The Lobe coerces Norm Abrahms into helping with his evil scheme using a block of precious wood and a nasty-looking, rusty, rugged chainsaw.
- And it wasn't even Norm's block of wood: it was one that The Lobe bought himself. Norm just has a deep respect for carpentry. And it really was a very nice block of wood.
- This is also how Arms Akimbo sells "Oops Insurance" - he moves around the room, destroys something and then goes "Oops" until the proprietor pays him off. Later in the episode, he apparently blows up a building this way.
A related trope shows its head in the Where on Earth Is Carmen Sandiego? episode "Scavenger Hunt" where Carmen correctly predicts that apparently chucking a Fabrage egg into the river where Zack and Ivy have cornered her will distract Ivy enough to let her make her escape. The player calls Carmen out on this afterward, and Carmen claims that since both Carmen and the player both knew that Ivy would undoubtedly save the egg it was never really in any danger.
The Simpsons parodies this, as always, with a private eye visiting Skinner. When he doesn't get any information, he reshuffles the pages on Skinner's desk. When Skinner blandly says he can reorganise them, the private eye uses a stapler. Cue the Big No.
- Homer himself attempts this during "Hungry, Hungry Homer". He razzes Marge's hairdresser by unscrewing caps and dropping things on the floor (only to get a confused reply from the hairdresser). He tries again to the owner of the local baseball team but is immediately intimidated into undoing his vandalism.
Kim Possible "Oh No, Yono", Monkeyfist manages to escape the museum by having his monkey ninjas pitch priceless pottery so that Kim and Ron have to catch them.
In Sonic the Hedgehog, Snively tortures Antoine by cooking escargot in margarine.