There was much more civil unrest during the 1960s than the 1950s. However, unrest in both decades was caused by the issues of African American rights and grievances. In the 1960s, but not the 1950s, unrest was also caused by the Vietnam War.
In the 1950s, there was little civil unrest in the US. The Montgomery Bus Boycott might be called civil unrest. So might the actions of anti-integration forces in Little Rock, Arkansas in 1957. Both of these came about because of segregation and the Civil Rights Movement.
In the 1960s, there was much more unrest and the unrest was much more serious. There was a great deal of violence against members of the Civil Rights Movement. There was violence, largely by African Americans, during the “long, hot summer” of 1967 and even more riots after Martin Luther King, Jr. was assassinated in 1968. There was the unrest in Chicago at the Democratic convention in that year. That unrest was largely as a result of the Vietnam War and the “counterculture.”