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We present a method for obtaining efficient probabilistic solutions to geostatistical and linear inverse problems in spherical geometry. Our Spherical Direct Sequential Simulation (SDSSIM) framework combines information from possibly noisy observations, that provide either point information on the model or are related to the model by a linear averaging kernel, and statistics derived from a-priori training models. It generates realizations from marginal posterior probability distributions of model parameters that are not limited to be Gaussian. We avoid the restriction to Cartesian geometry built into many existing geostatistical simulation codes, and work instead with grids in spherical geometry relevant to problems in Earth and Space sciences. We demonstrate our scheme using a synthetic example, showing that it produces realistic posterior realizations consistent with the known solution while fitting observations within their uncertainty and reproducing the model parameter distribution and covariance statistics of a-priori training models. Secondly, we present an application to real satellite observations, estimating the posterior probability distribution for the geomagnetic field at the core-mantle boundary. Our results reproduce well-known features of the core-mantle boundary magnetic field, and also allow probabilistic investigations of the magnetic field morphology. Small-length scale features in the posterior realizations are not determined by the observations but match the covariance statistics extracted from geodynamo simulation training models. The framework presented here represents a step towards more general approaches to probabilistic inversion in spherical geometry.