Cellular solids have numerous advantages over homogeneous materials in their mechanical properties, including high specific strength, specific stiffness, and energy absorption. The latter is of special importance in the automotive industry in the context of applications like crumple zones. In this work, we investigate the deep indentation of representative metallic cellular honeycombs using full-scale, ultra-high fidelity continuum finite element (FE) simulations. The simulations are carried out with the overarching goal of understanding the dependence of indentation response and energy absorption on the lattice type and lattice geometry. The simulations reveal field quantities such as effective plastic strain, strain rate, and velocity fields at high spatial and temporal resolutions. Our simulations also accurately capture the complex local elastic and plastic phenomena that accompany the indentation, including buckling, and inter- and self-contact of the cell walls. Interestingly, we observe two broad types of deformation patterns in our simulations, with implications for the relation between lattice geometry and energy absorption. In addition, the results shed light on the details of local plastic flow at different locations in the cellular solid, which can potentially aid in the rational design or tailoring of these lattices. Our work also reveals critical issues in the accurate modeling and simulation of these solids.