I remember there used to be a play button on my music files on my Desktop. I would click it, and it would play that song without opening anything up. Now I can't. I have to make do with opening Finder, switching to column mode, and then playing it. I don't want to use Quick Look, because it disappears when I move away. I may have switched view settings, updated Quicktime, etc. Do you know of a way that I can do this again?
I suspect you must have installed a very light weight media player at one time and associated your music files with it.
You should try VLC. It will play most audio files, and if you associate it with your audio files then it will play them with a double click as you had before. You can change the icon of VLC to a play button image, and then your associated music files will look like play buttons.