"On the Street Where You Live" is one of my go-to songs when I audition for a musical. I recall it being described as a song "so good you wonder why it's given to such a minor character" or something along those lines. That made me think about other musicals where standout songs are given to characters who don't factor much into the story.
Another that comes to mind is "Take What You've Got" from Kinky Boots the character who sings it only shows up for the one scene, and structurally it doesn't tell us much about Charlie we don't already know (although it does have a set up for "Everybody Say Yeah") and yet the song is a total banger.
Best Little Whorehouse in Texas does this rather heavily, with some of its most memorable songs going to its (sometimes barely) featured roles: “Doatsy Mae,” “24 Hours of Loving,” “The Sidestep,” “The Aggie Song.”
"...everyone finally shut up, and the audience could enjoy the beginning of the Anatevka Pogram in peace."
My immediate thought is “The Gentleman is a Dope” from Allegro, although several of the songs in that show also qualify.
Thinking on some of the above, it depends on how you define minor or insignificant. I think Freddy is a very significant character in MFL, but he doesn’t have a lot of stage time; same goes for Beth in Merrily. But relative to others, they’re minor characters. So it’s an interesting question that I think can yield a wide array of answers.
A few that come to mind (although the definition of minor/insignificant is certainly up for debate, especially since a minor character could be quite significant to the narrative):
A Sympathetic Ear - A Strange Loop Charming - Natasha, Pierre, and the Great Comet of 1812 Playing Nancy - Groundhog Day Turn it Off - The Book of Mormon The Joy You Feel - The Light in the Piazza I Know Where I've Been - Hairspray The Old Red Hills of Home - Parade The Acid Queen - Tommy The Miller's Son - A Little Night Music
I don't know if Fraulein Schneider in "Cabaret" qualifies as a minor/insignificant character. But when I first saw "Cabaret" in 1968 London with Judi Dench and Lila Kedrova, I was knocked back in my seat as Kedrova sang What Would You Do? A great song. More than half a century later, it still packs an emotional wallop for me. Here is Kedrova singing it.
Reading the above, I disagree with a few. For instance:
Fraulein Schneider has at least one more solo and one duet; Lotte Lenya was nominated for Best Actress in a Musical, so I would not consider that meeting the criteria set.
I would also question whether Carlotta is an insignificant character in Follies. Most of the roles in Follies are modest. I would agree with the poster re a number of the songs in Follies probably being reasonable proposals.
I don't disagree re The Miller's Son, although I have never understood the appeal of that song. I have always thought of it as filler, even though it has 'an' 11:00 o'clock slot in the show.
I would probably think of a several numbers from Nine, with Be Italian being the best 'match'. I could also argue that Lilliane had very little to do other than number, for which Liliane Montevecchi won a Tony.
It probably doesn't quite meet the bill, but the lion's share of I Have Dreamed is sung by the male, who is barely in the show.
Also maybe a little stretch, but Sit Down, You're Rocking the Boat comes close.
Another popular number from Gypsy whose charms have always escaped me is All I Need now Is the Girl.
Don't know if it qualifies as a great song, but The Grass is Always Greener seems to fit. Marilyn Cooper stopped the show and won a Tony for a role that probably involved 6 minutes of stage time.
I don't think any of the songs from Whorehouse remotely meet the 'great songs' requirement.
Agree re Not a Day Goes By, simply because of the timing of the song. Someone we have not seen previously gets arguably the best song in the show. Her role is a little larger, but when she actually sings the song, she is insignigicant. I remember being surprised at how little applause the number received relative to others in the show, but concluded that was because we had no emotional connection to the character, which is a (minor) gotcha with Merrily's structure.