Thanks GamerGirl and happy4ya. Your posts helped me resolve the same issue on my Galaxy SII with Android 2.3.6. On two occasions I had a single song play on its own, in entirety and at full volume even though the phone was on silent mode and no apps were listed as running. Lowering all volume settings also did not help.
I had just changed my microSDHC card a few days ago, going from 8GB to 32GB. I had also cleared my storage cache (to resolve a problem with duplicated playlists.) One or both of those changes seems to have caused a mix-up in notification sounds from my reMind app. Instead of playing the notification sounds I had previously set, a song got played as the notification instead.
Solution: go through all your notifications that are incorrectly playing a song and re-specify the sounds you want for them.