We present a unified analysis of the drag forces acting on oscillating bodies submerged in superfluid helium such as a vibrating wire resonator, tuning forks, a double-paddle oscillator, and a torsionally oscillating disk. We find that for high-Stokes-number oscillatory flows, the drag force originating from the normal component of superfluid helium exhibits a clearly defined universal scaling. Following classical fluid dynamics, we derive the universal scaling law and define relevant dimensionless parameters such as the Donnelly number. We verify this scaling experimentally using all of our oscillators in superfluid $4\mathrm{He}$ and validate the results by direct comparison with classical fluids. We use this approach to illustrate the transition from laminar to turbulent drag regime in superfluid oscillatory flows and compare the critical velocities associated to the production of quantized vortices in the superfluid component with the critical velocities for the classical instabilities occurring in the normal component. We show that depending on the temperature and geometry of the flow, either type of instability may occur first and we demonstrate their crossover due to the temperature dependence of the viscosity of the normal fluid. Our results have direct bearing on present investigations of superfluids using nanomechanical devices [Bradley et al., Sci. Rep. 7, 4876 (2017)].