  For various positive integers $n$, we show the existence of infinite families
of elliptic curves over $\mathbb{Q}$ with $n$-division fields,
$\mathbb{Q}(E[n])$, that are not monogenic, i.e., the ring of integers does not
admit a power integral basis. We parametrize some of these families explicitly.
Moreover, we show that every $E/\mathbb{Q}$ without CM has infinitely many
non-monogenic division fields. Our main technique combines a global description
of the Frobenius obtained by Duke and T\'oth with a simple algorithm based on
ideas of Dedekind.
