The laminar flow past rectangular prisms is studied in the space of length-to-height ratio ( $1 \leqslant L/H \leqslant 5$ ), width-to-height ratio ( $1.2 \leqslant W/H \leqslant 5$ ) and Reynolds number ( $Re \lessapprox 700$ ); $L$ and $W$ are the streamwise and cross-flow dimensions of the prisms. The primary bifurcation is investigated with linear stability analysis. For large $W/L$ , an oscillating mode breaks the top/bottom planar symmetry. For smaller $W/L$ , the flow becomes unstable to stationary perturbations and the wake experiences a static deflection, vertical for intermediate $W/L$ and horizontal for small $W/L$ . Weakly nonlinear analysis and nonlinear direct numerical simulations are used for $L/H = 5$ and larger $Re$ . For $W/H = 1.2$ and 2.25, the flow recovers the top/bottom planar symmetry but loses the left/right one, via supercritical and subcritical pitchfork bifurcations, respectively. For even larger $Re$ , the flow becomes unsteady and oscillates around either the deflected (small $W/H$ ) or the non-deflected (intermediate $W/H$ ) wake. For intermediate $W/H$ and $Re$ , a fully symmetric periodic regime is detected, with hairpin vortices shed from the top and bottom leading-edge (LE) shear layers; its triggering mechanism is discussed. At large $Re$ and for all $W/H$ , the flow approaches a chaotic state characterised by the superposition of different modes: shedding of hairpin vortices from the LE shear layers, and wake oscillations in the horizontal and vertical directions. In some portions of the parameter space the different modes synchronise, giving rise to periodic regimes also at relatively large $Re$ .