  Let $E\subseteq F$ and $E'\subseteq F'$ be Borel equivalence relations on the
standard Borel spaces $X$ and $Y$, respectively. The pair $(E,F)$ is
simultaneously Borel reducible to the pair $(E',F')$ if there is a Borel
function $f:X\to Y$ that is both a reduction from $E$ to $E'$ and a reduction
from $F$ to $F'$. Simultaneous Borel embeddings and isomorphisms are defined
analogously. We classify all pairs $E\subseteq F$ of smooth countable Borel
equivalence relations up to simultaneous Borel bireducibility and
biembeddability, and a significant portion of such pairs up to simultaneous
Borel isomorphism. We generalize Mauldin's notion of Borel parametrization in
order to identify large natural subclasses of pairs of smooth countable
equivalence relations and of singleton smooth (not necessarily countable)
equivalence relations for which the natural combinatorial isomorphism
invariants are complete, and we present counterexamples outside these
subclasses. Finally, we relate isomorphism of smooth equivalence relations and
of pairs of smooth countable equivalence relations to Borel equivalence of
Borel functions as discussed in Komisarski, Michalewski, and Milewski.
