The laminar—turbulent transition of the Taylor—Görtler (TG) vortex flow in the clearance between two concentric spheres with only the inner sphere rotating (spherical Couette flow) is investigated by velocity measurement and simultaneous spectral and flow-visualization measurements by measuring the intensity of light scattered by the aluminium flakes used in flow visualization in the case of a relatively small ratio of the clearance to inner-sphere radius (clearance ratio β = 0.14). An azimuthal velocity component has been measured by a constant-temperature hotwire anemometer at two different colatitudes (meridian angles) θ; θ = 80° and 90° (the equator). A critical Reynolds number, some transition Reynolds numbers, flow regimes and flow states are obtained by the simultaneous spectral and flow-visualization measurements. The flow state is expressed by the number of toroidal TG vortex cells N, that of spiral TG vortex pairs Sp, the wavenumber of the travelling azimuthal waves on the toroidal TG vortices m and the wavenumber of shear waves SH. The mean velocity distribution and the characteristic values of the fluctuating velocity, such as autocorrelation coefficient, power spectrum and turbulence intensity (r.m.s. value), are considered over a great range of Reynolds number Re. Three kinds of fundamental frequencies of the velocity fluctuation are discovered and their characteristics are clarified by means of the velocity measurement and the simultaneous spectral and flow-visualization measurements. The three kinds of fundamental frequencies expressed by fS, fW and fH correspond to the spiral TG vortices, the travelling azimuthal waves and the shear waves, respectively. These fundamental frequencies are independent of both θ and wall distances from the inner sphere, but depend strongly on Re. Although the rotation frequency of the travelling azimuthal waves (or wave speed) in the circular Couette flow decreases monotonically with increasing Reynolds number until it reaches a plateau, the values of the rotation frequencies of the spiral TG vortices, the travelling azimuthal waves and the shear waves in the spherical Couette flow, fS/SP, fW/m and fH/SH, are nearly constant as the Reynolds number is increased, and differ slightly from one another.