Context. Bright-rimmed clouds (BRCs) are isolated molecular clouds located on the edges of evolved HII regions. Star formation within these clouds may have been triggered through the propagation of photoionisation-induced shocks driven by the expansion of the HII region. Aims. The main focus of this paper is to investigate the current level of star formation within a sample of southern hemisphere BRCs and evaluate to what extent, if any, star formation may have been triggered. Methods. In this paper we present the results of a programme of position-switched CO observations towards 45 southern BRCs. The 12CO, 13CO and C18O J = 1-0 rotational transitions were simultaneously observed using the 22-m Mopra telescope. We complement these observations with archival mid-infrared data obtained from the MSX and Spitzer, as well as submillimetre and radio data previously reported in the literature. Combining all of the available data with the observations presented here allows us to build up a comprehensive picture of the current level of star formation activity within a significant number of BRCs. Results. Analysis of the CO, mid-infrared and radio data result in the clouds being divided into three distinct groups: a) clouds that appear to be relatively unaffected by the photoionisation from the nearby OB star(s); b) clouds that show evidence of significant interaction between the molecular material and the HII regions; c) clouds towards which no CO emission is detected, but are associated with strong ionisation fronts; these are thought to be examples of clouds undergoing an ionisation flash. We refer to these groups as spontaneous, triggered, and zapped clouds, respectively. Comparing the physical parameters of spontaneous and triggered samples we find striking differences in luminosity, surface temperature and column density with all three quantities significantly enhanced for the clouds considered to have been triggered. Furthermore, we find strong evidence for star formation within the triggered sample by way of methanol and H2O masers, embedded mid-infrared point sources and CO wings, however, we find evidence of ongoing star formation within only two of the spontaneous sample. Conclusions. We have used CO, mid-infrared and radio data to identify 24 of the 45 southern BRCs that are undergoing a strong interaction with their HII region. We can therefore exclude the other 21 sources (~50%) from future studies of triggered star formation. Fourteen of the 24 interacting BRCs are found to be associated with embedded mid-infrared point sources and we find strong evidence that these clouds are forming stars. The absence of mid-infrared sources towards the remaining ten clouds and the lack of any other evidence of star formation within these clouds leads us to conclude that these represent an earlier evolutionary stage of star formation.