I hate to state the obvious, but quite literally just select the song(s) you want to delete, and hit the DEL key on your keyboard, you will be prompted whether you want to continue and delete the song(s).
If that doesn't work, must be something wrong on your end because thats what I've done for years with no problem. Perhaps the playlist could be locked and you don't have permission to make changes, e.g. the playlist your trying to alter is from a different device.