This study provides strong evidence for a robust sequence of bifurcations transitioning from steady to quasiperiodic dynamics in the wake of compact and solid bluff bodies with an aspect ratio close to one. This sequence starts with spatial symmetry loss through a pitchfork bifurcation, then temporal symmetry break via a Hopf bifurcation, leading to periodic dynamics which is later affected by a Neimark-Sacker bifurcation, characterized by an eigenpair with an incommensurate frequency becoming unstable, thus giving rise to quasi-periodic dynamics. The proposed pitchfork/Hopf/Neimark-Sacker sequence is supported by a three-dimensional linear stability analysis on flow past a cube and a sphere to identify the first flow bifurcations past bluff bodies showing different symmetries in their geometry. Linear analysis is performed using a Newton-GMRES method. After confirming the pitchfork and Hopf bifurcations, it reveals fully 3D Floquet modes characteristic of the Neimark-Sacker bifurcation leading to quasi-periodic (QP) dynamics in both cases. These results therefore sustain the conjecture of a unified « pitchfork/Hopf/Neimark-Sacker » bifurcations sequence for compact bluff body flows. When considering the subsequent flow regimes, the following questions emerge: Which dynamics can be observed in bluff body flows between the QP-state and the apparition of temporal chaos ? Is the route to chaos identical in the cube and sphere cases ? To address these points, nonlinear simulations (DNS) trace the dynamics from quasi-periodicity to temporal chaos. The routes followed by the wake of the cube and the sphere are standard but clearly differ: the cube rather follows the "Feigenbaum's route" with frequency locking and a sequence period doublings prior to temporal chaos, while the sphere seem to follow the "Ruelle-Takens-Newhouse" route, with the introduction of a third frequency, which is expected to form a T3-torus and subsequently lead to chaos. A particular focus will be dedicated to this case.