Mesenchymal migration is a proteolytic and path generating strategy of individual cell motion inside the network of collagen fibres that compose the extracellular matrix of tissues. We analyze the spectral stability of the families of standing and traveling wave solutions of the one-dimensional version of the $M^5$-model, which was proposed by T. Hillen to describe mesenchymal cell movement. Regarding the standing waves, they are spectrally stable and the spectrum of the linearized operator around the waves consists solely of essential spectrum. To prove that in the standing case the point spectrum is empty we use energy estimates together with the integrated-variable technique of Goodman. The panorama is completely different in the traveling case; the wave profiles are spectrally unstable due to the fact that the essential spectrum reaches the closed right-half complex plane. In our pursuit of spectral stability, we have constructed a weighted Sobolev space where the essential spectrum lies inside the open left-half complex plane.