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Regularization te hniques for the numeri al solution of nonlinear inverse s attering problems in two spa e dimensions are dis ussed. Assuming that the boundary of a s atterer is its most prominent feature, we exploit as model the lass of artoon-like fun tions. Sin e fun tions in this lass are asymptoti ally optimally sparsely approximated by shearlet frames, we onsider shearlets as a means for the regularization in a Tikhonov method. We examine both dire tly the nonlinear problem and a linearized problem obtained by the Born approximation te hnique. As problem lasses we study the a ousti inverse s attering problem and the ele tromagneti inverse s attering problem. We show that this approa h introdu es a sparse regularization for the nonlinear setting and we present a result des ribing the behavior of the lo al regularity of a s atterer under linearization, whi h shows that the linearization does not a e t the sparsity of the problem. The analyti al results are illustrated by numeri al examples for the a ousti inverse s attering problem that highlight the e e tiveness of this approa h.