We consider systems of quasilinear partial differential equations of second order in two- and three-dimensional domains with corners and edges. The analysis is performed in weighted Sobolev spaces with attached asymptotics generated by the asymptotic behaviour of the solutions of the corresponding linearized problems near boundary singularities. Applying the Local Invertibility Theorem in these spaces we find conditions which guarantee existence of small solutions of the nonlinear problem having the same asymptotic behaviour as the solutions of the linearized problem. The main tools are multiplication theorems and properties of composition (Nemytskij) operators in weighted Sobolev spaces. As application of the general results a steady-state drift-diffusion system is explained.