In this article we analyze the following problem: given a mechanical system subject to (possibly redundant) bilateral and unilateral constraints with set-valued Coulomb's friction, provide conditions such that the state which consists of all contacts sticking in both tangential and normal directions, is solvable. The analysis uses complementarity problems, variational inequalities, and linear algebra, hence it provides criteria which are, in principle, numerically tractable. An algorithm and several illustrating examples are proposed.