What can a graduate student do in order to have a chance to stay permanently in USA upon finishing studies ?
There is no guaranteed way.
The most common path is this:
- Work on OPT after graduation (OPT is 12 months, can be extended to 29 months for STEM majors, during which you can work for a company in your field of study; there are certain restrictions, like you cannot have more than 90 days without work)
- Then find a company to sponsor you for H1b work visa during this time (H1b has an annual quota that runs out very quickly, which basically means it can only be applied for in April each year, and starts in October each year; in many years, there's a lottery, which means you may not get it even if a company is able and willing to sponsor you). H1b is for 3 years, and can be extended for 3 more years.
- Then get the company to (or move while on H1b to a company which is willing to) sponsor you for an employment-based green card before your H1b runs out.
Of course, there are other ways, like marrying a U.S. citizen, or asylum, or staying illegally.