Regarding the O1 Visa, both categories require outstanding achievements and extraordinary ability.
The evidence needed to demonstrate this will depend on your professional area. If you work in the television industry or in the performing arts, you may present evidence of a leading role in a motion picture or television production, proof of major commercial success, or positive critical reviews.
Of course, a major internationally recognized award, such as a Nobel prize, Academy award, Director’s Guild Award, or any other such award, will count strongly in your favor.
In the field of science, you must demonstrate authorship of scholarly articles and other published materials or of being employed in a critical role, especially if receiving substantial remuneration.
Regardless of your field of work, a specialized immigration lawyer will help you prove extraordinary achievement in your application.