The spherical Couette flow between two concentric spheres with only the inner sphere rotating is simulated by solving the 3D incompressible Navier-Stokes equations with a fifth order upwind compact finite difference method. Two moderate clearance ratios, β = (R2 − R1)/R1 = 0.14 and 0.18, respectively, are chosen for comparison with previous experimental and numerical results. First, the spiral Taylor-Görtler (TG) vortex flow at Re = 1110 for β = 0.14 [W. M. Sha and K. Nakabayashi, “On the structure and formation of spiral Taylor-Görtler vortices in spherical Couette flow,” J. Fluid Mech. 431, 323–345 (2001)10.1017/S0022112000003128] is found to develop traveling waves at Re = 1800. A wavy TG vortex flow formed at low Re numbers can return to steady TG vortex as Re number is increased to a critical value Re = 6600, thus confirming the occurrence of a reverse Hopf bifurcation from limit cycle to fixed point [K. Nakabayashi, W. M. Sha, and Y. Tsuchida, “Relaminarization phenomena and external-disturbance effects in spherical Couette flow,” J. Fluid Mech. 534, 327–350 (2005)]. Second, multiple supercritical flows for β = 0.18 [M. Wimmer, “Experiments on a viscous fluid flow between concentric rotating spheres,” J. Fluid Mech. 78, 317–335 (1976)10.1017/S0022112076002462] are simulated for a wide range of Re numbers from the first instability (Re ≈ 655) up to the proximity of transition to turbulence (Re ≈ 8000). The simulation confirms Wimmer's experimental observation that a periodic 2-vortex flow coexists with the steady 0- and 1-vortex flows in certain low Re range. There is also a reverse Hopf bifurcation for this periodic wavy 2-vortex flow at Re = 2270. As Re number is further increased, the steady 0- and 2-vortex flows begin to form spiral waves in the secondary flow region for Re ⩾ 6500, while the 1-vortex flow has similar spiral disturbances for Re = 8000. Multiple higher modes with different numbers of spiral waves can be generated by using different wavenumbers in the imposed perturbation. Detailed description of these multiple higher modes is given in terms of rotational frequency, wavenumber, and spatial structure.