This paper introduces an approach for flexible, robust Bayesian learning of structure in spherical data sets. Our method is based upon a recent construction called the needlet, which is a particular form of spherical wavelet with many favorable statistical and computational properties. We perform shrinkage and selection of needlet coefficients, focusing on two main alternatives: empirical-Bayes thresholding for selection, and the horseshoe prior for shrinkage. We study the performance of the proposed methodology both on simulated data and on a real data set involving the cosmic microwave background radiation. Horseshoe shrinkage of needlet coefficients is shown to yield the best overall performance against some common benchmarks.