A user of mine had an iPhone 5 that she was syncing with her iTunes library. She had removed everything from her library except a single album. After syncing the phone still had about 40 songs that would not go away. After multiple attempts at deleting them I found the easiest way to wipe all the music so we can start fresh with only the songs in her iTunes library.
On the phone open Settings > General > Usage and under STORAGE click Music, then Edit, then click the red circle with the line through it.