I would argue that the most controversial single event of this time was the Scopes "monkey" trial of 1925. This trial emphasized, and was caused by, the cultural divide between the America that was more rural and old-fashioned and the "new" America that was more identified with the cities and flappers and things like that.
A second of these events is the Teapot Dome scandal of 1923. This was a scandal in which a high government official (the Secretary of the Interior) took bribes in return for leasing federally owned land that had oil under it to the people who bribed him.
Finally, in 1927, Sacco and Vanzetti were executed. This was hugely controversial because many Americans felt that the men had been unjustly convicted based on the fact that they were immigrants who had radical political beliefs. Many people believed that they were actually innocent and were only being executed because of who they were.