Unsteady, three-dimensional computations and flow visualization experiments are carried out to study the onset of instabilities along the sidewall of a closed cylindrical container with a rotating lid. The computations simulate the impulsive spin-up from Re=3000 to Re=6000 for a cylinder of height-to-radius aspect ratio equal to 1.75. During the early stages of the spin-up, travelling Taylor-like ring vortices are shed from the rotating lid. The rings are initially axisymmetric but become azimuthally wavy as they approach the stationary cover. The appearance of the ring-mode is followed by the emergence of four pairs of quasi-stationary spiral vortices, which spread across the cylinder wall and cause the sidewall boundary layer to detach along spiral separation lines. Such lines are also detected in the laboratory flow visualization using aluminum paint pigment flakes. Visualization experiments are also carried out for Reynolds numbers higher than those simulated to further elucidate the evolution of the spiral mode and explore the onset of secondary instabilities.