In this experimental note, we consider the centrifugal instability of a laminar shear layer, generated by the impulsive start of the rotation of a circular solid cylinder about its vertical axis immersed in a linearly stratified fluid. The flow is determined by the Reynolds number, Re, based on the cylinder rotation rate and the cylinder radius, and the Froude number, Fr, represented by the ratio of the rotation frequency Ω over the buoyancy frequency N. The onset of the instability starts when the boundary layer reaches a certain thickness. We show for this boundary layer that there is a transition from the centrifugally unstable regime to a wave-like regime at Fr ≈ 1 and a stable flow below a critical Reynolds number. We focus on the centrifugally unstable regime Fr⪆1, for which the onset time and wavelength are predicted by scaling laws that depend on the Reynolds number. Agreement with the theoretical prediction of Kim and Choi [“The onset of instability in the flow induced by an impulsively started rotating cylinder,” Chem. Eng. Sci. 60, 599-608 (2005)] in a homogeneous fluid confirms that the instability of this boundary layer is not modified by the presence of stratification. These results therefore show that the centrifugal instability of the spin-up boundary is dominated by inertial motions, suggesting that close lateral boundaries, as in thin-gap stratified Taylor-Couette flow, increase the effects of buoyancy on the instability and wavelength.