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 $0.019.5$ and projected axis 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% of post-starburst galaxies at $9.510.5$); (2) at $9.510.5$, 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%). Our analysis suggests that AGN are `on' for $50%$ 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 $50%$ incomplete due to the exclusion of narrow-line AGN.

Accepted by MNRAS; 19 figures, 39 pages