Strictly speaking the Treadle is the pedal or platform for the foot or feet to rest on while rocking it to power a machine. The treadle is connected to the Pitman that drives the Band Wheel crank and turns the band wheel so that the belt powers the machine. The action of rocking the treadle by foot is to treadle.
Nowadays the word treadle commonly refers to the Treadle Stand which may be enclosed in a Cabinet Table, Parlor Cabinet, Drawing Room Cabinets etc. Singer examples may be found Singer Cabinets, Treadle Bases & Cases and Singer Cabinets & Cases