Oh I actually did a paper on this! The Supreme Court's main contribution to supporting the civil rights movement was the acts they passed in that area. Although the 60's had more to do with the rights of everyone in general, the 70's focused more on the education system. The busing act was extremely important because it allowed counties to change county guidelines to promote racial integration in public schools. The high school I graduated from was actually the first in my state and that's what I researched! This case is known as Swann v. Charlotte-Mecklenburg Board of Education, 402 U.S. 1 (1971) if you want more information. Gates v. Collier also ended legal racial segregation in prisons! Griggs v. Duke made it illegal to use intelligence tests as a requirement for African-Americans to get positions and Lau v. Nicolas made it unconstitutional for schools not to provide English translations to Chinese-speaking students.