A lot of songs that we associate with a specific movie, such as As Time Goes By, didn't debut in the film. This list is for those that did. Uploaded by images.fanpop.com.
This recognition isn’t for songs that were incorporated into a movie’s soundtrack, but for songs that made their debut in a film. Lots of wonderful choices, but here are my top ten.
No. 10, from Rocky, “Gonna Fly Now”
No. 9, from Saturday Night Fever, “Stayin’ Alive”
No. 8, from Toy Story 2, “When She Loved Me”
No. 7, from The Way We Were, “The Way We Were”
No. 6, from An American Tail, “Somewhere Out There”
No. 5, from The Happy Ending, “What Are You Doing the Rest of Your Life”
No. 4, from Shall We Dance, “They Can’t Take That Away from Me”
No. 3, from Swing Time, “The Way You Look Tonight”
No. 2, from Holiday Inn, “White Christmas”
No. 1, from The Wizard of Oz, “Somewhere Over the Rainbow”
Great American Things 
Outstanding selections – and you’re right about “Rainbow” being number 1. I’d like to include Astaire’s Putting on My Top hat, too, but don’t know what I bump to get it in.