The flow of pure He II at low temperatures and a range of pressures is probed using an electrostatically-driven oscillating grid. With increasing oscillation amplitude, a (history dependent) first threshold is reached where the initially pure superflow abruptly changes: the resonant frequency decreases and the response becomes strongly nonlinear, attributable to quantized vortices responding to the motion of the grid so as to increase its effective mass without additional damping. On further increase of oscillation amplitude a second threshold is reached, probably marking the onset of superfluid turbulence. The increase in effective mass is believed to be due to a boundary layer of vortex loops that can evolve into turbulent flow at the second threshold. Open questions and problems for future research are formulated.