Musical Gag

Simply put, a Musical Gag is either one of two things:

In music, a Musical Gag is a joke that comes from somewhere other than the lyrics.
In other media, a Musical Gag is a joke within the item's music/soundtrack.

The following tropes are Musical Gags of the second type:

The Day the Music Lied
Letting the Air Out of the Band
Musicalis Interruptus
- Record Needle Scratch (in most cases)
Sorry I Left the BGM On
Xylophone Gag (to an extent, anyway...)

To avoid excessive repetition, examples on this page should be of Musical Gags that don't already fit into the specific tropes above.

Examples of Musical Gag include:

Type 1

Music

Tom Lehrer, a satirist most famous in the 1950's and 1960's, had a distinctive style involving writing and performing songs poking fun at various (then-)current events. Most of the jokes are in the lyrics, so they would not be Musical Gags—but there was the occasional melodic joke. For example, at the end of "New Math", Lehrer quips: "Come back tomorrow night, we're gonna do... fractions." This is followed up by a quick melody on the piano which seems to translate as either "Oh, brother!" or "Just kidding!".
Peter Schickele, a composer who moonlights as a musical satirist, has created a number of compact discs purporting to follow the work of "PDQ Bach", a fictional classical composer. Every one of PDQ Bach's works is actually filled with as many Musical Gags as Schickele could fit in, be it including the melody of "Pop Goes the Weasel" or an Overly Long Gag consisting of going up and down the 5-note scale and adding in increasingly nonsensical instruments. Much to his dismay, these musical parodies have become so popular that they have eclipsed his serious work.
"Faerie's Aire And Death Waltz" could be considered a Musical Gag, in that even a layperson looking at the score should be easily able to tell that it's intentionally unplayable.
Filk Songs, in addition to the silly lyrics, will often include Musical Gags.
Mozart himself wrote a piece entitled A Musical Joke. There are a few wrong notes, but most of the joke is how the piece fails to develop (as if it were written by someone who was out of ideas).
Camille Saint-Saens included a number of musical gags in The Carnival of the Animals. For example, movement twelve, "Fossils", includes quotes from Twinkle Twinkle Little Star and the French nursery rhyme Au Clair de la Lune, among other musical "fossils".
The Led Zeppelin song "The Crunge" consists entirely of this trope. It is written to sound like James Brown style Funk, which is written to be danced to, but there are so many changes in the time signature that any attempt to actually dance to it would result in a faceplant.

Newspaper Comics

In Peanuts, Schroeder whistles Beethoven's F minor sonata to call Snoopy in; the Musical Gag is that this would require whistling two notes at once. Later strips have Woodstock whistling similarly complex melodies.

Radio

Hello Cheeky would have one or two humorous songs in each episode. One of them, I've Fallen For A Girl Called Agnes, is entirely about the fact that nothing rhymes with Agnes, and the rhythm of the song keeps changing in reference to the fact that the singer never finishes a verse.

Theatre

Chee-Chee, a famous musical flop by Rodgers and Hart based on the novel The Son of the Grand Eunuch, has a final scene where the main character is being taken away to have the relevant operation. The orchestra plays a few bars from The Nutcracker Suite.
- In one of Richard Rodgers' later works, the ballet Slaughter on Tenth Avenue, three cops on a raid enter to the tune of "Three Blind Mice."
In the musical Of Thee I Sing, the French soldiers enter to the first few bars of "An American in Paris." (Both were composed by George Gershwin.)
In The Mikado by Gilbert and Sullivan, when the Mikado's song mentions Bach as a punishment for the music hall singer, woodwinds play the theme from Bach's Fugue in G minor (BWV 542).

Type 2

Anime and Manga

In Suzumiya Haruhi, the Five-Man Band calls itself the "SOS Brigade". One of the image songs' beat is the Morse Code for "SOS".

Film

In Spaceballs, the pan across the Spaceballs' ship is made even funnier by the orchestra repeatedly trying to segue into the next part of the music only to find out that there's more of the ship to get past first.
In the movie Down With Love, a character claims that something doesn't ring a bell. The soundtrack contradicts him.

Live-Action TV

Much of the Source Music chosen for Lost was recorded by artists who died in plane crashes.

Multiple

On occasion, the Leitmotif for a Joke Character will itself be a Musical Gag.

Radio

One that requires some explanation: Hamish and Dougal was a show spun off from I'm Sorry I Haven't a Clue, featuring two of Clue‍'‍s regulars. In one episode, the rest of Clue‍'‍s cast join Hamish and Dougal as guest stars. After they've all been introduced, we hear a brief burst of Clue‍'‍s theme tune, The Shickel Shamble.
- Also, since Clue has a local pianist in the form of Colin Sell, he sometimes joins in on the joke-cracking. As one round of Just A Minim collapsed into shambles, he could be heard playing Enter The Gladiators (circus music), and you can sometimes hear him pausing as he's thrown off by one of the panellists or has to adjust his tempo.

Western Animation

South Park used a Leave It to Beaver-style soundtrack to make Cartman injecting Kyle with his own AIDS-infected blood sound like Wacky Hijinks.
In the Pinky and The Brain episode "It's Only A Paper World", the title characters attempt to Take Over the World by building a life-sized replica of the planet Earth out of paper-mâché and luring the population onto it so they can rule the real Earth without interference. All the music in the episode is based on themes from Dvorak's "New World" symphony.

Video Games

On the original Skies of Arcadia Dreamcast discs, if you inserted one in a CD player, instead of the standard "Insert Dreamcast discs into your Dreamcast not your CD player" message, one of three randomly selected custom messages to this effect issued by the voice actors for Vyse, Aika, and Fina would play.
In Castlevania: Symphony of the Night, Alucard would inform you that this disc only has game data. It would then play some music before going into the game data and being annoying static.