I took the ASVAB when I first enlisted in 1989. Where can I see what my scores were, even from that long ago?

your ERB should show it.

Mine are in my service record. If you have a copy or can get a copy of your record, they should be in there somewhere.

Should be on your DD214, you should be able to request a current copy. That said I have seen plenty of DD214's that don't have them or only have some of the scores.