  In this paper, we formulate the Drinfeld module analogue of a question raised
by Lang and studied by Katz on the existence of rational points on abelian
varieties over number fields. Given a maximal ideal $\fl$ of $\F_q[T]$, the
question essentially asks whether, up to isogeny, a Drinfeld module $\phi$ over
$\F_q(T)$ contains a rational $\fl$-torsion point if the reduction of $\phi$ at
almost all primes of $\F_q[T]$ contains a rational $\fl$-torsion point. Similar
to the case of abelian varieties, we show that the answer is positive if the
rank of the Drinfeld module is $2$, but negative if the rank is $3$. Moreover,
for rank $3$ Drinfeld modules we classify those cases where the answer is
positive.
