The Casimir effect for a scalar field in presence of delta-type potentials has been investigated for a long time in the case of surface delta functions, modelling semi-transparent boundaries. More recently Albeverio, Cacciappuoti, Cognola, Spreafico and Zerbini [7,8,47] have considered some configurations involving delta-type potentials concentrated at points of R^3; in particular, the case with an isolated point singularity at the origin can be formulated as a field theory on R^3 \ { 0 } with self-adjoint boundary conditions at the origin for the Laplacian. However, the above authors have discussed only global aspects of the Casimir effect, focusing their attention on the vacuum expectation value (VEV) of the total energy. In the present paper we analyze the local Casimir effect with a point delta-type potential, computing the renormalized VEV of the stress-energy tensor at any point of R^3 \ { 0 }; to this purpose we follow the zeta regularization approach, in the formulation already employed for different configurations in previous works of ours (see [25,26,27] and references therein).