Of course, I was over-thinking this, because the definition of a show tune is not as obvious as it might be. One might be tempted to count movie songs, but I am particularly prone to Bollywood earworms and those really should not count. (I am also very susceptible to Jimmy Buffett lately, but there is no way I can claim "Volcano" as anything remotely resembling a show tune.) At any rate, I will just list some things I think qualify. But I also feel obliged to point out that the songs I go around singing are not necessarily songs that are actually any good.
1) "Coffee In a Cardboard Cup." This is a Kander & Ebb song from 70 Girls, 70 and it is one of the two songs I tend to sing while making my morning coffee. (The other is "Java Jive," which did originate as a movie song but I can't really call a show tune.)
2) "Auto da Fe (What a Day)" from Bernstein's Candide. One would not think that the subject comes up frequently, but it's my immediate mental association with the phrase "what a day" which does come up a lot.
3) "Luck Be a Lady Tonight." This is, of course, one of Frank Loesser's most perfect songs and is from Guys and Dolls.
4) "Big D." Another Loesser song, in this case from "Most Happy Fella." The tendency of this song to get stuck in my head is one of the reasons I don't fly American through DFW more often.
5) "Nice Work If You Can Get It." The Gershwins wrote this for a movie originally, but it did get used in Crazy For You so I'll count it.
6) "What Can You Say in a Love Song?" I've really only known this song since March when I saw Life Begins at 8:40 at the Library of Congress. It's Harold Arlen's music, with lyrics by Ira Gershwin and Yip Harburg. That should be enough to recommend it.
7) "Chrysanthemum Tea." I seem to be the only person who loves the score of Pacific Overtures quite so much. I think "A Bowler Hat" is a better song (and I woke up this morning singing "Someone in a Tree" for some reason) but this is the Sondheim song I seem to sing most often. Possibly because I drink a lot of tea.
8) "Marry Me a Little" from Company. I could actually put most of that Sondheim score on this list, with the exception of "You Could Drive a Person Crazy" which drives me crazy.
9) "The Air Is Free." This is from Robert Lindsey-Nassif's The Flight of the Lawnchair Man. I make no claim that this is a brilliant song but it is fun.
10) "Soon It's Gonna Rain" from The Fantasticks by Tom Jones and Harvey Schmidt. I often sing this in response to reading the weather forecast.
11) "Vanilla Ice Cream" from Bock and Harnick's She Loves Me. This is just a lovely song from a show that doesn't always get the recognition it should.
12) "Oh to Be a Movie Star" from The Apple Tree. Another Bock and Harnick show that should be better known. I mostly like the start of the song ("Chimneys are cozy, chimneys are warm. I think of chimneys as ports in a storm.")
13) "Kesa" from See What I Wanna See by Michael John La Chiusa. I have no idea what triggers this. I do think La Chiusa is brilliant, but I have qualms about the deliberate shock in the lyrics.
14) "Blue Skies." This is one of the Irving Berlin songs I sing most often and a lot of my fondness for it has to do with an odd connection between its music and Jewish themes. At any rate, the other two Berlin songs I find myself singing a lot ("How Deep is the Ocean" and "I'll See You in C-U-B-A") are less likely to be legitimate show tunes.
15) "Make Believe" from Show Boat. Another classic I feel no need to justify.
I feel that there should be some Cole Porter song on this list and more Gershwin and something from Jerry Herman (who can write better music than he often does) and maybe something from Maury Yeston's Nine. But that would be overthinking it.