The transition from regular to Mach reflection remains a key area of study for unsteady oblique shock interactions with a solid surface. In this study, high-fidelity computational fluid dynamics (CFD) based on the combination of an upwind finite-volume spatial discretization and an anisotropic block-based adaptive mesh refinement (AMR) procedure applied to the solution of the Euler equations governing compressible gaseous flows is used in conjunction with a systematic post-processing methodology to accurately compute and compare the regular-to-Mach reflection (RR-MR) transition on wedges, cylinders, cones, and spheres. The objective of this study is to determine the similarities and differences that arise when oblique shock reflection is considered on cones, cylinders, and spheres relative to the well-studied case of a planar wedge using past theoretical, experimental and numerical results.