In this work we examined the effect of localized electrostatic potentials generated by subsurface nanostructures on overlaid graphene layers of various thicknesses and the screening produced by such layers. Electrostatic Force and Kelvin Probe microscopies were used to investigate few-layer graphene (FLG) domains on top of ionic crystals. Step edges, pits and protrusions within the ionic surface create sizeable and local perturbations of the surface potential of graphene overlayers. These were within the tenth of eV range in FLG with up to three layers, and become considerably screened in thicker layers. A computational Thomas-Fermi model of screening by graphene layers was developed to specifically describe localized potential distributions (not modelled previously), and correlated successfully with the experimental results presented here, results from literature, as well as available ab-initio Density Functional Theory results. Engineering such nanostructures in a regular manner can allow the bottom-up creation of on-sheet p-n junctions and superlattices that exploit the Dirac nature of carriers in graphene, and provide a test bed for studying local screening.