We perform atomistic simulations of dislocation nucleation in two-dimensional (2D) and three-dimensional (3D) defect-free hexagonal crystals during nanoindentation with circular (2D) or spherical (3D) indenters. The incipient embryo structure in the critical eigenmode of the mesoregions is analyzed to study homogeneous dislocation nucleation. The critical eigenmode or dislocation embryo is found to be localized along a line (or plane in 3D) of atoms with a lateral extent, $\xi $, at some depth, $Y\u2217$, below the surface. The lowest energy eigenmode for mesoregions of varying radius, r_{meso}, centered on the localized region of the critical eigenmode is computed. The energy of the lowest eigenmode, λ_{meso}, decays very rapidly with increasing r_{meso} and λ_{meso} ≈ 0 for $rmeso\u2273\xi $. The analysis of a mesoscale region in the material can reveal the presence of incipient instability even for $rmeso\u2272\xi $ but gives reasonable estimate for the energy and spatial extent of the critical mode only for $rmeso\u2273\xi $. When the mesoregion is not centered at the localized region, we show that the mesoregion should contain a critical part of the embryo (and not only the center of embryo) to reveal instability. This scenario indicates that homogeneous dislocation nucleation is a quasilocal phenomenon. Also, the critical eigenmode for the mesoscale region reveals instability much sooner than the full system eigenmode. We use mesoscale analysis to verify the scaling laws shown previously by Garg and Maloney in 2D [2016, “Universal Scaling Laws for Homogeneous Dissociation Nucleation During Nano-Indentation,” J. Mech. Phys. Solids, 95, pp. 742–754.] for the size, ξ, and depth from the surface, Y*, of the dislocation embryo with respect to indenter radius, R, in full 3D simulations.