This paper aims to give a mathematically rigorous description of the corner singularities of the weak solutions for the plane linearized elasticity system in a bounded planar domain with angular corner points on the boundary. The qualitative properties of the solution including its regularity depend crucially on these corner points or such types of boundary conditions. In particular, the resulting expansion of the solutions of the underlying problem involves singular vector functions, inlines, depending on a certain parameter ξ_{μ. We derive the transcendental equations for all ten possible cases of combinations of the boundary conditions generated by the basic four ones in classical elasticity proposing in the two natural directions of the boundary, i.e., tangential and normal direction, respectively, which depends on ξμ . So, a MATLAB program is developed whereby ξμ can be computed, and figures showing their distributions are presented. The leading singular exponents are computed for these combinations of the boundary conditions, wherein critical angles ω_ critical are listed such that for interior angles ω}