Critical exponents for the O(n) spin model local field near a surface, an edge and a corner are calculated by means of the renormalisation group method for the special geometry where the wedge or corner is formed by mutually perpendicular (d-1)-dimensional walls, each having von Neumann (special transition) or Dirichlet (ordinary transitions) boundary conditions. The combined effect of two or more interacting surfaces leads to new local susceptibility exponents which are calculated to order in ( in =4-d). The case of two or more von Neumann surfaces presents some difficulty in that a double divergence arises in the first order perturbation calculation of the local susceptibilities.