Abstract The authors use a nonlinear, compressible, spectral collocation code to examine the evolution and secondary instability of Kelvin–Helmholtz billows in stratified shear flows at intermediate Reynolds numbers. Two-dimensional results exhibit structure consistent with previous numerical studies and suggest dissipation via diffusive transports within the billow cores. Results obtained permitting three-dimensional structures show, in contrast, that secondary instability results in a series of counter-rotating vortices that occupy the outer portions of the billow structures, are oriented in the plane of two-dimensional motion, largely along the two-dimensional velocity field, and contribute substantially to mixing and homogenization of the billow cores at later times. Examination of the flow structure leading to secondary instability also suggests an alternative explanation of the nature of this instability in stratified flows to that offered previously. Comparison of the two-dimensional and spanwise-a...