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Carrier trapping is one of the main sources of performance degradation in nanocrystal-based devices. Yet the dynamics of this process is still unclear. We present a comprehensive investigation into the efficiency of hole transfer to a variety of trap sites located on the surface of the core or the shell or at the core/shell interface in CdSe nanocrystals with both organic and inorganic passivation, using the atomistic semiempirical pseudopotential approach. We separate the contribution of coupling strength and energetics in different systems and trap configurations, obtaining useful general guidelines for trapping rate engineering. We find that trapping can be extremely efficient in core-only systems, with trapping times orders of magnitude faster than radiative recombination. The presence of an inorganic shell can instead bring the trapping rates well below the typical radiative recombination rates observed in these systems.