The ZaP Flow Z-pinch plasma device at the University of Washington produces a small diameter (20-30 mm) dense Z-pinch plasma with typical electron density 1022-1023 m-3 and ion plus electron temperature 100-200 eV. The plasma is stable, with relatively low magnetic mode activity, for tens of microseconds. This is orders of magnitude longer than predicted by a simple ideal magnetohydrodynamic calculation. The probable stabilizing mechanism is radial shear in the axial plasma flow. The axially flowing Z-pinch is generated with a coaxial accelerator coupled to a pinch assembly chamber. After the pinch assembles a quiescent period occurs, during which the mode activity is significantly reduced. Multichord Doppler shift measurements of impurity lines show a large, sheared flow during the quiescent period and low, uniform flow profiles during periods of high mode activity. The plasma has a sheared axial flow that exceeds the theoretical threshold for stability during the quiescent period and is lower than the threshold during periods of high mode activity. The Z-pinch plasmas are globally stable for 700-2000 times the theoretically predicted kink growth time of a static Z-pinch. The end of the quiescent period corresponds to a decrease in acceleration of plasma and possibly suggests a means to extend the experiment to quasi-steady-state operation.