Three U.S. states originally had unicameral legislatures, Pennsylvania, Georgia, and Vermont. Unicameralism didn’t last long in those states. Georgia adopted a second chamber in 1789. Pennsylvania in 1790. Vermont held out a bit longer, adding a second chamber to its legislature in 1836. All states subsequently joining the Union, including Nebraska, came into it with bicameral legislatures.
Interest in unicameralism returned, however, with the advent of the Progressive movement but in new ways.