It's difficult to tell from the pic, but is that a Floyd? Locking tuners I assume because there's no locking nut?
Mine has no serial number that I can see, but I was able to date it based on some of the features that were only available between 1984 and 1986. I've been told that they often had the serial number on the jack plate (which is really weird) but I don't see one on mine.