The serial number on my '88 Spoiler was (almost) lost after a refret, a little bit of lemon oil made it faintly visible.
I think yours is from about the same time, so should have it under the 24th fret. If it's a bit newer, it could have been on the truss rods cover. Try to play a bit with light and you might see something in one of those locations.
There might be a production number under one of the pickups, with some luck Mica can link it to a serial number. (though that can take a while and is a bit of a long shot, did not work on my Spoiler)
Production number in my '88 Spoiler (88S4710)