We study the links between star formation history and structure for a large mass-selected galaxy sample at 0.05 < z_phot < 0.30. The galaxies inhabit a very broad range of environments, from cluster cores to the field. Using HST images, we quantify their structure following Hoyos et al. (2012), and divide them into disturbed and undisturbed. We also visually identify mergers. Additionally, we provide a quantitative measure of the degree of disturbance for each galaxy (roughness). The majority of elliptical and lenticular galaxies have relaxed structure, showing no signs of ongoing star formation. Structurally-disturbed galaxies, which tend to avoid the lowest-density regions, have higher star-formation activity and younger stellar populations than undisturbed systems. Cluster spirals with reduced/quenched star formation have somewhat less disturbed morphologies than spirals with normal star-formation activity, suggesting that these passive spirals have started their morphological transformation into S0s. Visually identified mergers and galaxies not identified as mergers but with similar roughness have similar specific star formation rates and stellar ages. The degree of enhanced star formation is thus linked to the degree of structural disturbance, regardless of whether it is caused by major mergers or not. This suggests that merging galaxies are not special in terms of their higher-than-normal star-formation activity. Any physical process that produces roughness, or regions of enhanced luminosity density, will increase the star-formation activity in a galaxy with similar efficiency. An alternative explanation is that star formation episodes increase the galaxies roughness similarly, regardless of whether they are merger-induced or not.