In the Ian McKellen/Judi Dench version of William Shakespeare's Macbeth, Ian McDiarmid completely steals the show with his Porter Speech. Shakespeare originally wrote the character for this type of performance.
The Ghost of King Hamlet is often done this way. Some reports suggest that the role was originally played by Shakey himself.
The First Gravedigger (or "Clown") in Hamlet was written this way because Shakespeare may have been trying to showcase his best comedic actor going head-to-head with his best dramatic actor.
The First Servant from King Lear appeared in only one scene just long enough to defend Gloucester and get killed, but has been called one of Shakespeare's most noble characters, and even has a poem written about him.
Richard Henry Lee's biggest contribution to American independence was proposing it to the continental congress, before he had to leave. This means the character based on him in 1776 had a small role. Nonetheless, his character and song "The Lees of Old Virginia" were memorable enough to win a Tony for actor Ron Holgate.
Much of Brian "Le Petit" Dewhurst's role in Cirque du Soleil's Mystere comes before it actually starts, as he handles a preshow while the ushers finish seating the audience; in the show proper he only appears in the opening sequence, a blackout skit, and a ten-or-so-minute setpiece in the final half-hour. (Usually in Cirque, a clown gets at least two in-show setpieces and often the preshow too; his role is smaller because one of the lead characters, Bebe Francois, overlaps with a clown act.) In fact, his character is a Screwy Squirrel who isn't "actually" part of the proceedings; he rarely appears in advertisements for it. But he is also a Cool Old Guy (in both the show and Real Life) who just about steals the show — and there is a lot to steal — with his antics and the bond he establishes with said audience from the preshow onwards.
In M. Butterfly, there is a two-scene sequence when one of the main characters picks up a debutante at an embassy party. In the second scene — which takes place the "morning after" their tryst — the debutante casually tells him "you have a nice weenie." When she sees he's uncomfortable with that particular pet name for a penis, she launches into a COMPLETELY HYSTERICAL monologue which distills the entire history of Western Civilization down into a dick-measuring contest, and then she walks offstage and is never seen again.
There are a number of instances of this in musical theatre: characters who come on for one quick scene which either contains or consists entirely of one big show-stopping number, and then never appear outside of the ensemble for the rest of the show.
The musical version of Footloose features Cowboy Bob, who sings the Irrelevant Act OpenerStill Rocking, which rocks, hits on Rusty (the secondary love interest), and is never heard from again. Bob still manages to get ridiculous applause.
Berthe in Pippin is one of the most classic examples of this.
Grandma in Billy Elliot is a very popular recent instance.
The Young Soldier in Parade is considered by some to be one of the best roles in the show, despite being onstage for only the first three minutes, and even that is only the first half of the first song.
As established by original Broadway production precedent, The Young Soldier is often doubled with another small role, Fiddlin' John.
Since the Donmar Warehouse Production, which brought a few substantial changes such as cutting/adding characters, scenes and songs; Fiddlin' John is no longer in the show. However, there is room for the Young Soldier to double with Frankie (a supporting role) as was done in the Donmar Production.
An argument could be made that the Proprietor in Assassins fits this trope.
Jonathan Freeman got himself a Tony nomination for playing the Headwaiter in She Loves Me. Enough said.
Eddie in The Rocky Horror Show.
Richard Henry Lee from 1776 would not technically qualify, due to having a little more to do than just that one scene and number. The embodiment of this trope, however, is Martha Jefferson. An actress who played Martha in a local production put it something like this: "I get to go on, make out with a handsome man for a little while, have a cute little scene, belt out a show-stopping song and chill backstage the whole rest of the time. That's a great gig."
Also noteworthy is the courier, who runs on and offstage every once in a while to deliver missives, but only has anything substantial to do during the (surprisingly low-key) final scene/song of Act One, "Momma, Look Sharp".
The Teen Angel in Grease. In Broadway and touring productions, Stunt Casting is often used for this role (the 2009 U.S. tour cast Taylor Hicks from American Idol, for example); Frankie Avalon played him in the movie version.
In Show Boat, Joe does very little except sing "Ol' Man River" at the end of the first scene. (One advertisement for the original production bills his actor only as the singer of "Ol' Man River," while crediting the other featured players by their characters' names.) He actually also appears in a few later scenes, but those are mostly excuses for him to reprise the song. When Paul Robeson played Joe in the 1936 film version, an extra song was written for him.
Nimue in Camelot. May appear onstage or sing her one song from the wings, depending on the production.
The trio of strippers in Gypsy who explain to Louise why "You Gotta Get A Gimmick."
Daddy Brubeck and the other "Rhythm of Life" Church members in Sweet Charity. Also doubles as an Irrelevant Act Opener.
Steve in Paint Your Wagon. He does nothing at all, except sing the show's big hit song.
The title role in The Mikado. He has the top billing in the show, yet he's only in three songs in Act 2.
Don Attilio in the "Il Muto" scene from The Phantom of the Opera basically exists just for the atmosphere, yet he often nets one of the biggest laughs in the show by holding a very long, very low note at the end of one of his recitatives. (Note that the actor in this role usually doubles on one or two other minor parts as well.)
And then in the movie, they have the nerve to change it to Piangi playing the part. And Piangi being a tenor, kiss that low note goodbye.
The Phantom himself to an extent. Despite being the most complex and interesting character in the musical by a huge margin—not to mention being the titular character, he's only onstage for about forty minutes of its two-and-a-half hour running time. Of course, his seemingly supernatural abilities and established Scooby-Doo Hoax help him retain an invisible yet palpable presence throughout the entire production.
Pharaoh in Joseph and the Amazing Technicolor Dreamcoat. One scene, one song, a solo spot in the curtain call with usually the most applause of anyone in the cast.
Usually the role is doubled with Potiphar (who only appears in act one). Of course to do this, an actor has to do an English Music Hall Song, and then do Elvis (that's pretty awesome).
Still on Andrew Lloyd Webber territory: the evicted mistress of Juan Peron in Evita, which does nothing but being insulted by Eva, sing "Another suitcase in another hall" (which was designed from the start to be a show-stopping ballad) and leave. The level of dissonance between the importance of the character and the success of the song is such, that for the movie version the poor girl has to make do with singing two lines from the chorus and the song is retconned into being sung in full earlier by Madonna in the title role.
Follies is full of these: Stella and "Who's that Woman", Heidi and "One More Kiss", Hattie and "Broadway Baby", but above all, Carlotta with "I'm Still Here". Yvonne De Carlo was the original Carlotta, and frequently the most famous star in a Follies cast will be the Carlotta.
Evil Dead: The Musical features the character of Ed, who is constantly interrupted, and only gets in one or two words at a time until the number "Bit Part Demon" outside of this, however, Ed is indeed, a glorified extra.
The Cradle Will Rock, a Mark Blitzstein musical, has Ella Hammer, the sister of a worker who died in the steel factory, sing only one song, and hers is the song everyone is talking about as they're leaving the theatre.
Melba has really one purpose in Pal Joey: to appear in a scene which ends with her singing (and sort of stripping to) "Zip!" This was a small enough part that Elaine Stritch could be playing it in New Hampshire and simultaneously be on call as Ethel Merman's understudy in another show on Broadway.
The priest in Man of La Mancha gets one of the best songs, "To Each His Dulcinea", and then fades back into the crowd of inmates/actors. He's also usually played by a very crazy, mute inmate.
Likewise, the prostitute Saraghina in Nine doesn't have much to do but gets to sing "Be Italian," by far the most memorable number in the show..
In the film version, this role was brilliantly given to Fergie from the Black-Eyed Peas.
The irrelevant comic relief characters Dick and Mae appear twice in Street Scene, but their second appearance hardly counts. Their first appearance was not so interesting in the original play, which didn't give them the show-stopping number "Moon-Faced, Starry-Eyed."
The character Marge MacDougall from the musical Promises, Promises is introduced near the beginning of the second act, gets one duet with the male lead, has almost no importance plot-wise, and exits the story almost as abruptly as she came in, never to be heard from again. However, the character is so popular that both of the actresses who potrayed her on Broadway won Tony Awards for Best Featured (Supporting) Actress.
Do you like my owl?
Mama from Memphis may qualify for her showstopping "Change Don't Come Easy" song. Though she is a frequent character and sings more than once, "Change Don't Come Easy" particularly stands out.
Herbert in Tanz der Vampire. He has one line in the first act, appears wordlessly (except for some singing over a backing chorus) in the second scene of the second act, proceeds to have a showcase song/scene full of Ho Yay with the hero that is generally regarded as one of the funniest (or sexiest, if you like that sort of thing) parts of the whole show a few scenes later, and then is demoted to harmonizing on two lines with his father at the ball and with Magda in the finale, and yet he is probably at least the second- if not THE- most popular character in the show, with one reviewer commenting that it's hard not to squee when he shows up, even if you don't like the actor playing him. The actor doesn't even double in the ensemble scenes before Herbert's entrance.
Vindication: This article- from the producers of the show- calls Herbert Tanz's "arguably the most popular figure".
Harvey Johnson, the nerdy kid from the opening number of Bye Bye Birdie, is easily the most memorable character from the whole play, despite having about three lines.
Pretty much any of the numbers from Cats count, with the exception of those involving Grizzabella.
Rusty Charlie in Guys and Dolls. He seems to be stuffed into the opening song with Benny and Nicely only then to disappear and not have any lines or songs the rest of the show.
Big Julie. He's basically there to steal the entire sewer craps game scene out from under Sky and Nathan - and succeeds, depending on how good your Sky's "Luck Be a Lady" is.
The very beautiful aria "Di rigori amato" from "Der Rosenkavalier" by Richard Strauss is sung by a character called "An Italian Singer" who comes in, sings the aria, is interrupted and leaves, not to be seen again.