  Let $E$ be a vector bundle over a smooth curve $C$, and $S = \mathbb{P} E$
the associated projective bundle. We describe the inflectional loci of certain
projective models $\psi \colon S \dashrightarrow \mathbb{P}^n$ in terms of Quot
schemes of $E$. This gives a geometric characterisation of the Segre invariant
$s_1 (E)$, which leads to new geometric criteria for semistability and
cohomological stability of bundles over $C$. We also use these ideas to show
that for general enough $S$ and $\psi$, the inflectional loci are all of the
expected dimension. An auxiliary result, valid for a general subvariety of
$\mathbb{P}^n$, is that under mild hypotheses, the inflectional loci associated
to a projection from a general centre are of the expected dimension.
