Nonlinear kink instabilities of high-Reynolds-number supersonic shear layers have been studied using high-resolution computer simulations with the piecewise-parabolic-method (PPM). The transition region between the two fluids of the shear layer is spread out over many computational zones to avoid numerical effects introduced on the smallest lengthscales. Mach number, density contrast, and perturbation speed and amplitude were varied to study their effects on the growth of the kink instabilities. In response to a perturbing sound wave, a travelling kink mode grows in amplitude until enough of a disturbance on the shear layer has been created for it to roll up and rapidly grow in thickness. The time it takes for this rapid growth to be initiated is proportional to the initial shear-layer thickness and increases for increasing Mach number or decreasing perturbation amplitude. For equal density, Mach 4 shear layers, perturbed by a sound wave with a 2% amplitude at the travelling mode velocity, the growth time is τg = (546 ± 24) δ/c, where c is the sound speed and δ the half-width of the shear layer.