If a college graduate or doctoral student is dismissed from school, what legal options do they have?

There may be several legal options available depending upon whether or not the school is a public school or a private school. Most schools have some sort of internal appeal process. That process should be exhausted. If it is a public school, you also then have a right to file what's called a writ of certiorari to a circuit court. If it is a private school, one option that you have is always to sue the school if you believe that the school failed to do something or did something improper.