Heroes Unleashed: The Next Generation
Magic doesn’t really exist. Most supers are mutants. However, science has grown by leaps and bounds, so powers explained by super-science is fine. There are also normal human heroes getting by on their wits and luck. Also, no alien races have yet contacted humans.
Characters will start with 180 power points. This is negotiable for characters returning from the first game, although remember, with great power comes great responsibility. And I might one-shot your character for no good reason.
1988 in this setting is not too different than the real 1988. Events, famous people, entertainment, etc. are mostly the same.