The Keller-Segel model for bacterial chemotaxis supports travelling wave solutions which have been described in the literature as both linearly stable and unstable and in the case of linear consumption (conditionally) nonlinearly stable. We reconcile this apparent contradiction by locating the essential spectrum, absolute spectrum and point spectrum of the linear operators associated with the travelling wave solutions. We derive conditions for the spectral (in)stability of the travelling wave solutions and the critical parameters that indicate a transition from a transient to absolute instability. Furthermore, we show that the absolute spectrum deforms as the consumption is changed, illustrating a connection between the constant, sublinear and linear cases.