The magnetostrophic regime, in which Lorentz and Coriolis forces are in balance, has been investigated in a rapidly rotating spherical Couette flow experiment. The spherical shell is filled with liquid sodium and permeated by a strong imposed dipolar magnetic field. Azimuthally travelling hydromagnetic waves have been put in evidence through a detailed analysis of electric potential differences measured on the outer sphere, and their properties have been determined. Several types of waves have been identified depending on the relative rotation rates of the inner and outer spheres: they differ by their dispersion relation and by their selection of azimuthal wavenumbers. In addition, these waves constitute the largest contribution to the observed fluctuations, and all of them travel in the retrograde direction in the frame of reference bound to the fluid. We identify these waves as magneto-inertial waves by virtue of the close proximity of the magnetic and inertial characteristic time scales of relevance in our experiment.