Post-starburst galaxies can be identified via the presence of prominent Hydrogen Balmer absorption lines in their spectra. We present a comprehensive study of the origin of strong Balmer lines in a volume-limited sample of 189 galaxies with 0.01<z<0.05, log(M*/M_{sun}_)>9.5 and projected axial ratio b/a>0.32. We explore their structural properties, environments, emission lines, and star formation histories, and compare them to control samples of star-forming and quiescent galaxies, and simulated galaxy mergers. Excluding contaminants, in which the strong Balmer lines are most likely caused by dust-star geometry, we find evidence for three different pathways through the post-starburst phase, with most events occurring in intermediate-density environments: (1) a significant disruptive event, such as a gas-rich major merger, causing a starburst and growth of a spheroidal component, followed by quenching of the star formation (70 per cent of post-starburst galaxies at 9.5<log(M*/M_{sun}_)<10.5 and 60 per cent at log(M*/M_{sun}_)>10.5); (2) at 9.5<log(M*/M_{sun}_)<10.5, stochastic star formation in blue-sequence galaxies, causing a weak burst and subsequent return to the blue sequence (30 per cent); (3) at log(M*/M_{sun}_)>10.5, cyclic evolution of quiescent galaxies which gradually move towards the high-mass end of the red sequence through weak starbursts, possibly as a result of a merger with a smaller gas-rich companion (40 per cent). Our analysis suggests that active galactic nuclei (AGNs) are 'on' for 50 per cent of the duration of the post-starburst phase, meaning that traditional samples of post-starburst galaxies with strict emission-line cuts will be at least 50 per cent incomplete due to the exclusion of narrow-line AGNs.