We propose a class of spherical wavelet bases for the analysis of geophysical models and for the tomographic inversion of global seismic data. Its multiresolution character allows for modelling with an effective spatial resolution that varies with position within the Earth. Our procedure is numerically efficient and can be implemented with parallel computing. We discuss two possible types of discrete wavelet transforms in the angular dimension of the cubed sphere. We describe benefits and drawbacks of these constructions and apply them to analyse the information in two published seismic wave speed models of the mantle, using the statistics of wavelet coefficients across scales. The localization and sparsity properties of wavelet bases allow finding a sparse solution to inverse problems by iterative minimization of a combination of the ℓ2 norm of the data residuals and the ℓ1 norm of the model wavelet coefficients. By validation with realistic synthetic experiments we illustrate the likely gains from our new approach in future inversions of finite-frequency seismic data. © 2011 The Authors Geophysical Journal International © 2011 RAS.