  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 $0.01<z<0.05$, $\log(\mbox{M}_{\star}/\mbox{M}_{\odot})>9.5$ 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 $9.5<\log(\mbox{M}_{\star}/\mbox{M}_{\odot})<10.5$ and 60% at
$\log(\mbox{M}_{\star}/\mbox{M}_{\odot})>10.5$); (2) at
$9.5<\log(\mbox{M}_{\star}/\mbox{M}_{\odot})<10.5$, stochastic star formation
in blue-sequence galaxies, causing a weak burst and subsequent return to the
blue sequence (30%); (3) at $\log(\mbox{M}_{\star}/\mbox{M}_{\odot})>10.5$,
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%). 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.
