  Let $q\geq2$ be a prime power and consider Drinfeld modules of rank 2 over
$\mathbb{F}_q[T]$. We prove that there are no points with coordinates being
Drinfeld singular moduli, on a family of hyperbolas $XY=\gamma$, where $\gamma$
is a polynomial of small degree. This is an effective Andr\'e-Oort theorem for
these curves. We also prove that there are at most finitely many Drinfeld
singular moduli that are algebraic units, for every fixed $q\geq2$, and we give
an effective bound on the discriminant of such singular moduli. We give in an
appendix an inseparability criterion for values of some classical modular
forms, generalising an argument used in the proof of our first result.
