We present an analysis of the mass flow rates of cooling flows in an X-ray flux-limited ample of clusters of galaxies, in which most of the clusters have been observed with both imaging and spectroscopic detectors. Using high-quality images and constraints from broad-band spectra, we find that the fraction of clusters with central cooling times less than the Hubble time is high (at least 70 per cent and possibly 90 per cent), indicating that cooling flows are a common and long-lived phenomenon. The fraction is higher than found in previous studies, due to our consideration of the effects of the spatial resolution and the signal-to-noise ratio of the images. We also determine the fraction of cooling flows with optical line emission and the space density of clusters as a function of the mass flow rate. The clusters that are not cooling flows are apparently recent mergers (e.g. A2256) and contain radio halo sources. We investigate the impact of cluster evolution through mergers on the evolution of the mass flow rate in four well- studied cooling flows, and estimate the length of time for which they have been undisturbed.