Applying a sufficiently rapid start–stop to the outer cylinder of the Taylor–Couette system, structures approximately aligned with the rotation axis were recorded in the classic work of Coles (1965 J. Fluid Mech. vol. 21, no. 3, pp. 385–425). These short-lived rolls are oriented perpendicular to the classic Taylor-vortex rolls. In this work we report numerical observation of this instability, guided by a more recent experimental observation. The instability is shown to be related to an inflection in the azimuthal velocity profile, a finding consistent with the experimental observations of its emergence during the deceleration phase. Despite the transient nature of start–stop experiments, we show that the instability can be linked to that of the oscillating boundary layer problem of Stokes. There are several reasons why the instability may have remained elusive, both for experimental observation and for the idealised system. We look in more detail at dependence on the radius ratio for the Taylor–Couette system, $\eta=R_i/R_o$, where $R_i$ and $R_o$ are the inner and outer radii. We find that, in the case where the size of the rolls scales with the gap width, for radius ratios any lower than that used by Coles, $\eta=0.874$, the instability is quickly overrun by axisymmetric rolls of Görtler type.