I noticed this problem about half a month ago and thought the new 6.1 Update would fix it for my iPhone 5. I listen to my music through playlists, generally 15-30 songs long. The playlists are set to shuffle and continuous playing.
I've noticed that whichever song is selected as the first track of the random playlist, it is skipped over and replaced by another random song in the playlist when the playlist finishes and loops back around to play the first song again. I've also noticed that if I skip forward then backwards, the original first song in the playlist is back at the number one spot.
I've tried both turning the phone on and off, closing the multitasking off, and restoring the phone to factory settings. Nothing seems to work.