The rule concerning cores is that if you kill a Robot (not mechas) with a super-effective move, you have a 50% of getting their core from them. For example, If you finished Fire Man off with an Ice Slasher, you'd have a 50% of getting his Flame Core from him.

What this means is that if no robots in the game have, say, a Laser Core, then you can't get a Laser Core as of now. From your list, you CAN get a:

Crystal Core from Gemini Man, Jewel Man, or Crystal Man
Copy Core from Mega Man, Bass, Proto Man
Neutral Core from Disco, Rhythm, Roll, Enker, Punk, or Ballade.
I may have missed some robots...
Happy hunting!

You can also get Space Cores from Star Man.

aaaaand that's about it. No Laser or Shield Cores are available, because--as Doc said--there's no Laser- or Shield-type RM's in the game, yet.

Neutral: Roll, Rhythm, Disco, Enker, Punk, or Ballade. (Quint not included. lol)
Copy: Mega Man, Proto Man, or Bass.
Shield: Wait for King or Shield Man to be released, then go to them.
Laser: You'll have to fire your lazor for the core when Laser Man or Terra is around.
Crystal: Gemini Man, Crystal Man, and Jewel Man.
Space: Star Man. (And Ring Man, apparently. Unless Mikey/Mike is wrong, but idk)

You're welcome. I think I should make a core hunt guide...

I keep thinking of his Ring Boomerang being both Space and Cutter types. xD

Post edited. Thanks, Tails.