In this paper we study newly developed methods for linear elasticity on polyhedral meshes. Our emphasis is on applications of the methods to geological models. Models of subsurface, and in particular sedimentary rocks, naturally lead to general polyhedral meshes. Numerical methods which can directly handle such representation are highly desirable. Many of the numerical challenges in simulation of subsurface applications come from the lack of robustness and accuracy of numerical methods in the case of highly distorted grids. In this paper we investigate and compare the Multi-Point Stress Approximation (MPSA) and the Virtual Element Method (VEM) with regards to grid features that are frequently seen in geological models and likely to lead to a lack of accuracy of the methods. In particular we look how the methods perform near the incompressible limit. This work shows that both methods are promising for flexible modeling of subsurface mechanics.