Weak shock reflection at wedge angles near glancing incidence with respect to the reflecting surface exhibits interesting structures such as the three wave von Neumann reflection (vNR), four-wave Guderley and Vasi’lev reflections (GR & VR) due to the von Neumann paradox. In this study, weak conical shock waves at small wedge angles ( \(\theta _w = 5^{\circ }\) ) are investigated numerically and theoretically in the steady flow regime. Through inviscid numerical simulations, we find that neither the GR nor VR appears in the planar and axisymmetric regimes, contrary to the shock polar predictions. Instead, both cases clearly demonstrate the appearance of the vNR structure, with only minor differences between each other. Simple theoretical models are applied based on small perturbations of the Riemann invariants in inviscid conical flow, which agree reasonably well with results obtained from numerical simulations, while discrepancies exist near the centreline, due possibly to the appearance of the paradoxical reflection structures.