  For a class $\mathcal K$ of countable relational structures, a countable
Borel equivalence relation $E$ is said to be $\mathcal K$-structurable if there
is a Borel way to put a structure in $\mathcal K$ on each $E$-equivalence
class. We study in this paper the global structure of the classes of $\mathcal
K$-structurable equivalence relations for various $\mathcal K$. We show that
$\mathcal K$-structurability interacts well with several kinds of Borel
homomorphisms and reductions commonly used in the classification of countable
Borel equivalence relations. We consider the poset of classes of $\mathcal
K$-structurable equivalence relations for various $\mathcal K$, under
inclusion, and show that it is a distributive lattice; this implies that the
Borel reducibility preordering among countable Borel equivalence relations
contains a large sublattice. Finally, we consider the effect on $\mathcal
K$-structurability of various model-theoretic properties of $\mathcal K$. In
particular, we characterize the $\mathcal K$ such that every $\mathcal
K$-structurable equivalence relation is smooth, answering a question of Marks.
