  A quasi-hereditary algebra is an Artin algebra together with a partial order
on its set of isomorphism classes of simple modules which satisfies certain
conditions. In this article we investigate all the possible choices that yield
to quasi-hereditary structures on a given algebra, in particular we introduce
and study what we call the poset of quasi-hereditary structures. Our techniques
involve certain quiver decompositions and idempotent reductions. For a path
algebra of Dynkin type $\mathbb{A}$, we provide a full classification of its
quasi-hereditary structures. For types $\mathbb{D}$ and $\mathbb{E}$, we give a
counting method for the number of quasi-hereditary structures. In the case of a
hereditary incidence algebra, we present a necessary and sufficient condition
for its poset of quasi-hereditary structures to be a lattice.
