Westminster Chimes

"doo dee doo doo... doo dee DEE doo... dee doo dee doo... dee dee DEE doo..."

The most common tune played by clock chimes, in both in Real Life and fiction, is the one known as Westminster Quarters. The tune is constructed from five different sequences combining four pitches. Not all five are played at once: even at the full hour, only the last four are used, followed by between one and twelve strokes on a deeper bell. The most popular of these sequences would seem to be the fourth (mi, do, re, sol) and fifth (sol, re, mi, do). (These are also played at third quarter, followed by the first.) Therefore, the most commonly heard in fiction are the fourth and fifth sequences, or the fourth alone, often repeated for effect in either case.

Japanese school bells often sound like this, to the point where it's a Stock Sound Effect in high school anime.

A somewhat common variant is to use only the fourth sequence, but with the first note raised a half step so that the first two notes make a falling fourth rather than a falling major third. In this augmented variation, the tonic is usually transposed to the first note from the second (i.e. do, sol, la, re rather than fa, do, re, sol).

A little-known fact is that the quarters have lyrics. There are many variations, but the lyrics used by Big Ben (featured on a plaque in the clockworks room) are:

All through this hour
Lord be my guide
And by Thy power
No foot shall slide

Examples of Westminster Chimes include:

Anime and Manga

Used in one of the Tenchi Muyo! movies as part of an Establishing Shot for a school.
In K-On!, the school uses a synthesized Westminster Quarters to mark the end of class periods.

Film

Incorporated in the score for Mary Poppins, during the rooftop scene, between orchestral reprises of "Spoonful of Sugar" and "Feed the Birds".
Used during the climactic scene of Sherlock Holmes.
The doorbell at the home of the eponymous Laura in Laura.
In The Ghost and Mr. Chicken, when Luther Heggs is spending the night in the haunted house, these bells can be heard in the distance just before the famous scene with the self-playing organ and the bleeding portrait.

Literature

In Reaper Man, a demon-powered pocketwatch announces the half-hour by saying "Bing bing bong bing. Bingely-bingely bong bing bing". This trope is so familiar that even in print you can tell it's meant to be Westminster Quarters.

Live-Action TV

The Theme Song to Yes Minister was based on this motif.
Scrubs: "Wrong, wrong, wrong, wrong... wrong, wrong, wrong, wrong."
The Beverly Hillbillies had a doorbell that made this sound. Naturally, whenever someone rang the doorbell, they had no idea where the sound was coming from and attempted to search for the song's source. Their search was always then interrupted by someone knocking on the door.
The Prisoner: The chimes are a key plot point in "The Chimes of Big Ben". They help No. 6 realise he hasn't escaped to London, and remains a captive in The Village.
The King of Queens: The doorbell of Doug and Carrie's annoying neighbors, the Sackskys.

Music

The intro to "Let 'Em In" by Wings.
- Paul wasn't the only ex-Beatle to work the Westminster Chimes into one of his solo songs. They also feature quite prominently as a Recurring Riff in George Harrison's "Ding Dong, Ding Dong."
Played on the harp in the "London Symphony" by Ralph Vaughan Williams.
The Fugue in A Major from "The Short-Tempered Clavier" by PDQ Bach uses this complete with twelve o'clock chime.
The opening to The Ohio State University Marching Band's rendition of the school's alma mater, Carmen Ohio, has the trumpet sections mimicking the bells (which play on campus every fifteen minutes at the Orton Hall belltower.)
The jazz standard If I Were a Bell starts out with a jazzy version of the chimes.
Jeff Beck quotes this in The Yardbirds song "Jeff's Boogie".
The opening riff of X's "Los Angeles" is meant as a variation on this - it's done at a fast enough tempo that you might not catch the intentional similarity at first.
The intro of Do Not Forsake Me Oh My Darling's "The Chimes Of Big Ben". Which makes sense because the song is inspired by the already mentioned Prisoner episode of the same name.
Supertramp uses the chimes during the one of the collage sections in "Fool's Overture".

Theatre

"Tender Shepherd" in the musical Peter Pan, though the low "sol" is often avoided by the singers.

Video Games

This forms the basis for the background music of Twinkle Elementary in MOTHER 1 and Sunflower Elementary in Hamtaro: Ham-Hams Unite!
Used as the background music for the pinball stage in Parodius Da!
It also appears briefly at the beginning of the Tokimeki Memorial stage in Jikkyou Oshaberi Parodius.
- And by extension, it's a staple of the Tokimeki Memorial series, being Konami's Japanese High School Romance Game series.
The fourth segment is used at the beginning of the Clefairy pointing minigame in Pokémon Stadium.
- Pokémon Stadium 2 uses it for battles in Earl's Academy—the chimes play, then it segues into an arrangement of the Pokemon Center theme.
Persona 4 plays the fourth segment at the end of every school day in-game, naturally it's a Japanese school you're at.
Nine Hours Nine Persons Nine Doors uses this as a puzzle- You have to play it on a piano that has its keys rearranged.
Used as menu music in Grand Theft Auto: London 1969.
Parodied in Dawn of War, where an Ork says the word "Orkz" in this fashion.
In Thwaite, after each round, one of the sequences plays to let the player know an hour has passed.

Web Original

The Wrestlecrap Radio podcast briefly featured a device called the Clocktrolla (meant to count down the days until WWE Women's Champion Candice Michelle eclipsed The Fabulous Moolah's twenty-eight-year reign as champion); upon activation, the Clocktrolla played a terrible rendition of the tune, performed on steam whistles.

Western Animation

Used for Rarity's doorbell in My Little Pony: Friendship Is Magic.

Truth in Television

Well the Trope Namer and Trope Codifier is Big Ben isn't it?
Lots of doorbells have this as one of their standard digital tones.
- Some electronic air raid sirens have it as a test signal. So imagine the electronic doorbell tones, but at earsplitting volume, too!
Just like mentioned above, Japanese Schoolbells.
Many antique clocks used this as a chime after "Big Ben" was built. However, the tune really originated from Great St. Mary's church in Cambridge, England, and was known as the "Cambridge" chimes before it was used in "Big Ben" at the Westminster Palace.
Brownies (The UK equivalent of Girl Scouts) often used to end meetings with a song to this tune: Oh Lord our God/ Thy children call/ Grant us thy peace/ And bless us all. Nowadays, because of the reference to God, another song is often used instead.
Played during the daylight hours by the carillon in the Peace Tower on Parliament Hill in Ottawa, Canada -- except at the eleventh hour of the eleventh day of the eleventh month.

Examples of the augmented variation

Theatre

Begins each iteration of "Poor Professor Higgins" in My Fair Lady.
The play-out of "If I Were A Bell" from Guys and Dolls.
The final cadence of "Everything's Coming Up Roses" in Gypsy.