  We show that a positive proportion of cubic fields are not monogenic, despite
having no local obstruction to being monogenic. Our proof involves the
comparison of $2$-descent and $3$-descent in a certain family of Mordell curves
$E_k \colon y^2 = x^3 + k$. As a by-product of our methods, we show that, for
every $r \geq 0$, a positive proportion of curves $E_k$ have Tate--Shafarevich
group with $3$-rank at least $r$.
