In 1967's Loving v. Virginia decision, the Supreme Court ruled that laws banning interracial marriages were unconstitutional. Chief Justice Earl Warren wrote for the unanimous court that "Marriage is one of the 'basic civil rights of man,' [...] There is patently no legitimate overriding purpose independent of invidious racial discrimination which justifies this classification. The fact that Virginia prohibits only interracial marriages involving white persons demonstrates that the racial classifications must stand on their own justification, as measures designed to maintain White Supremacy." In 1968, a Gallup poll found that just 20 percent supported, while 72 percent disapproved, of the thought that blacks and whites should be able to get married.

There's also the Warren Court's banning of prayer in public school in Engel v. Vitale, 370 U.S. 421 (1962). Gallup showed that at the time, 79% of Americans approved of religious observances in public schools.

A handful of the Warren Court decisions on the free exercise clause were controversial at the time. One National Review columnist went so far as to say the decision made the United States the only country outside of the Iron Curtain that curtailed religion in schools. (Don't have the source for this, but you'd probably be able to find it.)

Don't forget Brown v. Board of Education of Topeka (Brown II). Although Brown I is arguably more famous, Brown II was the decision that actually ordered desegregation to proceed with “all deliberate speed"—a requirement that sparked "massive resistance" from Southern states, particularly Virginia. The Virginia General Assembly went so far as to completely shut down public school districts in order to avoid complying with the Brown II mandate—a practice that wouldn't be struck down until 1964 in Griffin v. School Bd. of Prince Edward Cty.

Even after these extreme anti-desegregation measures were struck down, the Court would spend years invalidating the "creative" solutions of school districts committed to trying to circumvent Brown II. Only in 1968, in Green v. County School Board did the Court strike down so-called “freedom of choice” plans, which permitted minority students to enroll in white schools only if they made an affirmative request to do so.

It took almost fifteen years after Brown II for the Court to expend the political capital necessary to hold that school districts have an “affirmative duty” to eliminate racial discrimination “root and branch.” So while I don't have direct polling data, Brown II was nonetheless deeply unpopular throughout large swaths of the South.

Other sources: Chermerinsky's "Constitutional Law," my Constitutional Law class at the University of Texas School of Law

Keeping up with the Warren Court love, Brown v. Board of Education was enough of a departure from public opinion that when the case came back to the Court a second time, it instructed the school district to desegregate "with all deliberate speed"; i.e. take your time, we know it's a touchy, complicated subject. I recall hearing a statistic in law school that even after a decade, the vast majority (90%?) of segregated schools had yet to desegregate.

Another example could be Dred Scott. It repealed the Missouri Compromise and was a large factor in starting the Civil War. I'm not entirely sure what was the public opinion regarding slavery at the time, but the North was a lot more populous than the South (though the South had a lot of political power from counting slaves as 3/5 of a person), and the upstart Republican Party was elected into the Presidency and Congress a mere 4 years later.

Also, during the early to mid 1930s, the Supreme Court issued a number of controversial decisions overturning progressive legislation, such as New York's minimum wage law in Morehead v. New York. These decisions were at the very least deeply unpopular with the mostly Democratic Congress and, of course, the Democratic President. FDR and Congress threatened to increase the number of justices on the Supreme Court in response.