We use the Planck Early Release Compact Source Catalogue combined with Wilkinson Microwave Anisotropy Probe (WMAP) and other archival measurements to construct continuum spectra of three nearby dusty star-forming galaxies: Messier 82, NGC 253 and NGC 4945. We carry out a least-squares fit to the spectra using a combination of simple synchrotron, free–free and thermal dust models, and look for evidence of anomalous microwave emission (AME). We find that the radio spectra of all three galaxies are consistent with steep spectrum synchrotron emission, with a significant amount of free–free emission required to explain the Planck and WMAP data points in the frequency range 30–150 GHz. This brings the star formation rate based on free–free emission into better agreement with that from the non-thermal emission. We place limits on the presence of AME in these galaxies, finding that it is lower than expectations based on the ratio of far-infrared to AME from the Galaxy. Nevertheless, the shape of the spectrum of NGC 4945 hints at the presence of AME with a peak around 30 GHz. Future Planck data will let us look more closely at these galaxies, as well as to extend the analysis to many more galaxies.