I can answer because I had an O1 visa, and from what I've understood the equivalent green card has roughly the same requirements. If I wanted to apply for a green card while on my O1 visa, I could have went to the same process (I chose to move back to Europe instead). Interestingly going from an O1 visa to the equivalent green cards doesn't require a corporate sponsorship.
Basically if you have something like a Nobel price or a Turing award, you don't need anything else to get the O1 (and probably the green card). If you're like most people and don't have this kind of award, you still have a shot but you need to gather as much evidence as possible. From what my attorney told me at the time, weight matters.
That includes, for science:
- publications
- recommendation letters (preferably a blend from business, research and politicians)
- awards (even things like "the best paper" of a small conference with only 30 papers from local researchers)
- interviews
- be a member of an association only open to certain persons (for example with a Ph.D. you qualify for IEEE membership, and they will deliver a good-looking membership certificate)
So having a Ph.D is not enough, but if are a decently successful researcher and have a good network for the recommendation letters (or your future employer), you can get it without being of Turing award level.
The "extraordinary ability" is applicable also for athletics or business, but I don't know much about those.