Effects of radially and vertically sheared mean azimuthal flow on trapped waves at a seamount are shown to include changes to resonance frequency and spatial structure and the formation of two types of critical surface. In waves calculated using bathymetry, stratification strength, and frequency (0.66ƒ) appropriate to amplified diurnal currents observed at Cobb Seamount (46.8°N), mean flow based on measurements there does not cause critical surfaces. Even though clockwise mean flow vorticity is a significant fraction of ƒ, effects on the wave are limited to weakly reduced spatial scales and an increase in resonance frequency much weaker than estimates of the Doppler shift. The Doppler shift is opposed by variations in mean horizontal shear that cause a downslope potential vorticity gradient opposite that of the bottom slope supporting the wave; mean vertical shear is unimportant. In the general case, mean flow can cause (1) a stratified seamount-trapped wave critical surface where the intrinsic frequency vanishes, causing both horizontal and vertical scales of the wave to diminish and (2) an internal wave critical surface where the intrinsic frequency rises to the effective Coriolis frequency, or low-frequency bound for superinertial internal waves. An internal wave critical surface bounds a “superinertial cap” within which subinertial currents are effectively superinertial. Mean flow at Fieberling Guyot (32.4°N) reduces the effective Coriolis frequency by 0.43ƒ and causes a superinertial cap having an internal wave critical surface for diurnal currents (0.93ƒ) that coincides with measured turbulence maxima extending several kilometers away laterally above the seamount.