Relevant to my interest, will be monitoring this thread intently.
I'm a bit picky when it comes to scoring systems; infinite loops or a counterstop is usually an instant disqualification for me. Thus far, I've stumbled upon precious few I would compare favorably to your average post 4th-gen shmup. Here's the main ones that stand out to me:
Psycho Soldier (1987 auto-scrolling run 'n' gun, by SNK. Mainly based around letting the point items float to the top of the screen, then grabbing them at the last possible second, as items accrue in value the longer they stay on-screen. Figuring out how to route everything and stay alive simultaneously is quite tricky, you'll often have to exploit your bomb attack's invulnerability frames to manage this.)
Psychic 5 (1987 platformer, by NMK. The most insanely convoluted arcade scoring system ever conceived up until this point. I can't do it justice, read
this and scroll down to the sections pertaining to all the Bonuses.)
Gussun Oyoyo (1993 action puzzle game, by Irem. Bizarre mashup of Tetris & Lemmings, high scores are heavily reliant on speeding through levels to attain the maximum time bonus of 50000 points. It gets interesting due to the tetromino [well,
tromino in this case] drops being randomized, so you can't rely on receiving the same loadout every time; each playthrough is a unique experience. Quick thinking & adaptability is key. There's also a multiplier when you crush multiple enemies at once, but it rarely comes into play outside of the penultimate stage, where in typical Irem fashion you're encouraged to checkpoint milk. Even in their
puzzlers it's inescapable.

)
Denjin Makai (1994 belt scroller, by WinkySoft. I go into the nitty gritty over
here, read that if you must. TL;DR stay at low health the entire game, use your Desperation Move to finish off enemies whenever possible, guess correctly during the one or two sections with random point item drops.)
Pretty Soldier Sailor Moon (1995 belt scroller, by Gazelle. Toss every enemy like a ragdoll, milk certain bosses with your infinite combo, don't bomb, get lucky with randomized bomb drops, and somehow do all this while
not getting dicked over by the rudest timer in b'mup history.)
BTW, were we keeping this exclusive to arcade games? If not, lump in the following:
Elemental Gearbolt (1997 light gun shooter, by Alfa System. Based around maintaining your chain combo, never getting hit, strategic swapping between your Thunder shot & your Blaze shot, and in the US version, finding hidden chalices.)
The Adventure of Little Ralph (1999 platformer, by New Corporation. An elaborate fruit chaining system, an
extremely satisfying combo mechanic that has you domino effect-ing enemies into each other, the 10K point items you have to grab on a precise frame... a near masterpiece of scoreplay, marred only by an
unfortunate oversight that requires you milk arrows for an hour straight, if you want anything near a WR tier score.)