This isn't very common. Since the Rodgers and Hammerstein era, most songs are written for a specific character or story, and so a song that is written for one show isn't usually appropriate for the next one. And even when musicals were looser and songs were written less for character and more to fill designated "spots" (comedy number, belt, eleven o'clocker, ballad), there wasn't a huge amount of recycling, because every show had a different set of performers, and song that was written for, say, Ethel Merman couldn't be dusted off and given to Gertrude Lawrence.
Still, there have always been individual songs that a songwriter or producer couldn't quite bring himself to give up on. The most famous example is George and Ira Gershwin's "The Man I Love." They wrote it for Adele Astaire in Lady, Be Good!; it was cut. George Gershwin, knowing that the song would be a sure-fire hit if only he could get it into a show, kept on trying to fit it in somewhere. He got it into Strike Up the Band in 1927, but the show closed out of town (it would eventually reach Broadway three years later, but without "The Man I Love"). So he tried to get Marilyn Miller to sing it in Rosalie in 1928, and again it was cut. Finally Gershwin gave up and let it be published as a stand-alone song, and it became a huge hit.
An example of a cut song that actually did make it into another project is "Say a Prayer For Me Tonight," written for My Fair Lady. Eliza Doolittle (Julie Andrews) was supposed to sing the song before going off to test her newfound training at the Embassy Ball. It was cut, and quite rightly so; it slowed the show down, and Andrews already had plenty of numbers to sing. But two years later, Vincente Minnelli had Lerner and Loewe pull the song out of cold storage and gave it to the title character of Gigi. But that demonstrates the danger of re-using cut songs. Not only is "Say a Prayer" one of the weakest scenes in the movie (in my opinion anyway), the song, having been written for one character, is just wrong for this other character: for example, it's a bit incongruous for a French girl to sing "Pray I'll be Wellington, not Bonaparte." (Yes, Wellington won, but still. Gigi shouldn't be unpatriotic.)
Then there's the case of a really good song that gets cut from a hit show, and gets incorporated into a flop. This can't be fun for the songwriters, who are left with the knowledge that the song would have gotten famous if only it had managed to survive as part of the hit show. A song that comes to mind is "Smashing New York Times," which Charles Strouse and Lee Adams wrote for Applause. It was a musical adaptation of All About Eve, and the song was an ironic ballad about one of the themes of the source material -- theatre people who are more interested in their careers than in their emotional lives:
We trade our lives
For just some good reviews,
For "smashing" - New York Times,
"Terrific" - Daily News.
We do our shows,
And what does it all mean?
We're grateful when we get
"Just swell" - Time Magazine.
The words we need are not from our lover
But words some stranger writes.
From newspaper presses
Won't warm you on winter nights.
So hold me close
And whisper soft to me:
"Perfection" -- CBS,
"Just lovely" -- NBC.
And as we part,
These words I need the most:
"Come back soon" -- New York Times,
"I love you" -- New York Post.
It was by some distance the best and most interesting song Strouse and Adams wrote for Applause, so of course it got cut. Applause was probably the worst hit musical of its era, if not of all time -- with talented people like Strouse, Adams, Betty Comden and Adolph Green contributing some of the poorest work of their careers -- so it figured that the best song would get chopped. But Strouse and Adams, rightly proud of the song, found a place for it seven years later when they wrote A Broadway Musical, another back-stager (based loosely on their experiences writing the Sammy Davis Jr. vehicle Golden Boy). Since this show was also about theatre people, "Smashing New York Times" fit fairly well into it, even though ultimately it would make more sense in the context of the All About Eve story. But the show, despite a pretty good score, was a bomb, lasting something like one night. And "Smashing New York Times" wasn't heard again until Jason Graae sang it on one of Kimmel's "Lost in Boston" albums.
More trunk songs in a later post.