We quantify the amplitude of the lopsidedness, the azimuthal angular asymmetry index and the concentration of star-forming regions, as represented by the distribution of the Hα emission, in a sample of 78 late-type irregular galaxies. We bin the observed galaxies into two groups representing blue compact galaxies (BCDs) and low-surface-brightness dwarf galaxies (LSBs). The light distribution is analysed with a novel algorithm, which allows detection of details in the light distribution pattern. We find that while the asymmetry of the underlying continuum light, representing the older stellar generations, is relatively small, the Hα emission is very asymmetric and is correlated in position angle with the continuum light. We show that the concentration of continuum light is correlated with the Hα concentration; this implies that the young star formation has the same spatial properties as the older stellar populations, but that these properties are more strongly expressed by the young stars. We test a model of random star formation over the extent of a galaxy by simulating H ii regions in artificial dwarf galaxies. A galaxy is traced by assuming red star clusters distributed on an underlying exponential disc of radius twice the scalelength. The disc is allowed to change in apparent magnitude, scaleradius, position angle and ellipticity. We compare the asymmetry-concentration distribution predicted by the simulations with the real observed distribution; we find that only LSBs match the distribution predicted by the model. The reason is that, independently of the number of H ii regions, LSBs show no particular location of H ii regions, whereas BCDs show current star formation activity restricted very much to the central parts of the galaxies. A consideration of the properties of the continuum light leads to the conclusion that most of LSBs can be approximated by exponential discs of radius twice their scalelength; BCDs call, however, for much more concentrated underlying systems, with smaller scalelengths than assumed in the simulations. The implication is that random star formation over the full extent of a galaxy may be generated in LSB dwarf irregular galaxies but not in BCD galaxies.