  Let $E$ be an elliptic curve over $\mathbb{Q}$. Greenberg has posed a
question whether the structure of the fine Selmer group over the cyclotomic
$\mathbb{Z}_p$-extension of $\mathbb{Q}$ can be described by cyclotomic
polynomials in a certain precise manner. A recent work of Lei has made progress
on this problem by proving that the fine Mordell-Weil group (in the sense of
Wuthrich) does have this required property. The goal of this paper is study the
analogous question of Greenberg over various $\mathbb{Z}_p$-extensions of an
imaginary quadratic field $F$. In particular, when the elliptic curve has
complex multiplication by the ring of integers of the imaginary quadratic
field, we obtain analogous results of Lei over the cyclotomic
$\mathbb{Z}_p$-extension and anti-cyclotomic $\mathbb{Z}_p$-extension of $F$.
In the event that the elliptic curve has good ordinary reduction at the prime
$p$, we further obtain a result over the $\mathbb{Z}_p$-extension of $F$
unramified outside precisely one of the prime of $F$ above $p$. Finally, we
study the situation of an elliptic curve over the anticyclotomic
$\mathbb{Z}_p$-extension under the generalized Heegner hypothesis. Along the
way, we establish an analogous result for the BDP-Selmer group which may be of
independent interest. Motivated by our results, we pose questions analogous to
that of Greenberg in each of the above settings.
