A surface-wave phase-velocity inversion scheme based upon a compactly supported cubic B-spline basis on a triangular grid of knots with approximately equal inter-knot spacing is implemented to perform global, regional and variable resolution inversions. Numerical results show that the inverted phase-velocity maps based upon this method are similar to those obtained using a global spherical-harmonic representation. Nonlinear inversions incorporating ray refraction are feasible based upon the spherical-spline model parameterization and the resulting maps are only slightly different from those based upon the great-circle approximation. A multiresolution analysis is introduced to obtain finer resolution in regions with good ray coverage and lower resolution in regions with poor coverage.