We present a realistic application of an inversion scheme for global seismic tomography that uses as prior information the sparsity of a solution, defined as having few nonzero coefficients under the action of a linear transformation. In this paper, the sparsifying transform is a wavelet transform. We use an accelerated iterative soft‐thresholding algorithm for a regularization strategy, which produces sparse models in the wavelet domain. The approach and scheme we present may be of use for preserving sharp edges in a tomographic reconstruction and minimizing the number of features in the solution warranted by the data. The method is tested on a data set of time delays for finite‐frequency tomography using the USArray network, the first application in global seismic tomography to real data. The approach presented should also be suitable for other imaging problems. From a comparison with a more traditional inversion using damping and smoothing constraints, we show that (1) we generally retrieve similar features, (2) fewer nonzero coefficients under a properly chosen representation (such as wavelets) are needed to explain the data at the same level of root‐mean‐square misfit, (3) the model is sparse or compressible in the wavelet domain, and (4) we do not need to construct a heterogeneous mesh to capture the available resolution.