This paper is devoted to describing representations of the plane elastostatics operators in orthonormal bases of compactly supported wavelets. We are interested in giving an alternative numerical model to the finite element method and Fourier analysis, because wavelets bases are well-suited to adaptivity and discontinuities. The discretization of the elastostatics operators leads to matrices where elementary terms are easily computed from integrals of two or three wavelets and their derivatives. These integrals are performed as eigenvectors of low order matrices and given in filter banks. Mathematical results are worked out in details. A numerical example is presented which demonstrates the efficiency of the representation and shows it being consistent with analytical solutions.