The linear parallel and incompressible stability of a family of bluff‐body wake profiles is studied at Reynolds numbers close to the onset of Ka´rma´n vortex shedding. The family of mean flow profiles allows for the variation of the wake depth as well as for a variable ratio of wake width to mixing layer thickness. The absolute or convective nature of the sinuous instability is determined as a function of the profile parameters and Reynolds number. A comparison of this survey with experimental data shows that in bluff‐body near wakes a region of local absolute instability begins to form at a Reynolds number of approximately one‐half the critical value for Ka´rma´n vortex shedding. Hence, at the onset of the global response (Ka´rma´n vortex shedding), a substantial region of local absolute instability already exists in the wake. This confirms the qualitative model prediction of Chomaz, Huerre, and Redekopp [submitted to Phys. Rev. Lett.] and also shows that the prediction of vortex shedding frequencies, when based on local stability properties alone, is somewhat arbitrary evenatthe critical Reynolds number.