We consider the viscous and resistive dissipation of perpendicularly polarized shear waves propagating within a planar magnetic X-point. To highlight the role played by the two-dimensional geometry, the damping of travelling Alfvèn waves that propagate within an unbounded, but non-orthogonal X-point topology is analyzed. It is shown that the separatrix geometry affects both the dissipation time and the visco-resistive scaling of the energy decay. Our main focus, however, is on developing a theoretical description of standing wave dissipation for orthogonal, line-tied X-points. A combination of numerical and analytic treatments confirms that phase mixing provides a very effective mechanism for dissipating the wave energy. We show that wave decay comprises two main phases, an initial rapid decay followed by slower eigenmode evolution, both of which are only weakly dependent on the visco-resistive damping coefficients.