Three-dimensional (3-D) wake transition for flow past a square cylinder aligned with sides perpendicular and parallel to the approaching flow is investigated using direct numerical simulation. The secondary wake instability, namely a Mode A instability, occurs at a Reynolds number (
) of 165.7. A gradual wake transition from Mode A* (i.e. Mode A with vortex dislocations) to Mode B is observed over a range of
from 185 to 210, within which the probability of occurrence of vortex dislocations decreases monotonically with increasing
. The characteristics of the Strouhal–Reynolds number relationship are analysed. At the onset of Mode A*, a sudden drop of the 3-D Strouhal number from its two-dimensional counterpart is observed, which is due to the subcritical nature of the Mode A* instability. A continuous 3-D Strouhal–Reynolds number curve is observed over the mode swapping regime, since Mode A* and Mode B have extremely close vortex shedding frequencies and therefore only a single merged peak is observed in the frequency spectrum. The existence of hysteresis for the Mode A and Mode B wake instabilities is examined. The unconfined Mode A and Mode B wake instabilities are hysteretic and non-hysteretic, respectively. However, a spanwise confined Mode A could be non-hysteretic. It is proposed that the existence of hysteresis at a wake instability can be identified by examining the sudden/gradual variation of the 3-D flow properties at the onset of the wake instability, with sudden and gradual variations corresponding to hysteretic (subcritical) and non-hysteretic (supercritical) flows, respectively.