The song that won the "Best Song" Oscar was the one played in the music store, shortly after the characters first meet, and the first time she has access to a piano.

The song used for the studio-recording scene, "When Your Mind's Made Up," is probably my favorite in the film, the one I would have nominated for the Oscar competition. It would certainly have been just as likely to win. Maybe it was held back from consideration because it's written in "difficult" 5/4 time. The director's DVD commentary points out that the other three actor/musicians sen in the studio are not actually heard on this song, that the song is actually being played by the Frames (Glen's regular band) plus Marketa. The young drummer who appears onscreen, in particular, was singled out for not being able to play well enough on this the song. I don't imagine many (or any) of the other songs were overdubbed in this manner; the low budget cannot have allowed for very much such manipulation.

There's another concurrent thread about "Once," by the way, initiated back when the film first came out and revived right now thanks to the Academy Award connection...