An exhaustive parametric study of the transition scenario in the wake of oblate spheroids and flat cylinders placed with their rotation axis parallel to the flow is presented. The flatness of the investigated objects is classified by the aspect ratio χ defined as χ = d/a for spheroids (with d the diameter and a the length of the polar axis) and as χ = d/h) for cylinders (with h the cylinder height). We find a significant qualitative similarity between both configurations. At large aspect ratios (χ > 2.3 for spheroids and χ ≥ 4 for cylinders), the secondary bifurcation giving rise to a periodic state without planar symmetry is subcritical with a hysteresis interval of about two Reynolds number units. For spheroids, the sphere-like scenario is recovered only at aspect ratios very close to one (χ ≥ 1 are considered), while for cylindrical bodies the same holds for χ ≤ 1.7. For intermediate aspect ratios, a domain of states with non-zero net helicity separates states typical for the sphere wake from those of an infinitely flat disk.