The aim of this talk is to provide a characterization for sparse solutions of variational inverse problems with finite dimensional data. We consider the minimization of functionals that are the sum of two terms: a convex regularizer and a finite dimensional soft constraint. It was observed for specific examples that minimizers of variational problems of this type are sparse in a suitable sense. We formalise this fact proving the existence of a minimizer that is represented as a finite linear combination of extremal points of the unit ball of the regularizer. This finding provides a natural notion of sparsity for abstract variational inverse problems. We apply our result to the case of the TV regularizer. In this case, we characterize the extremal points of the BV seminorm for bounded domains justifying the staircase effect typical of the TV regularization in image denoising. We then consider the framework of dynamic inverse problems with the Benamou-Brenier energy as a regularizer. Using the classical theory of Optimal Transport, we are able to apply our abstract result in this specific case characterizing the sparse solutions. This allows to construct an algorithm converging to a sparse solution.