  Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a
number field $F$ with good supersingular reduction at all primes above $p$. We
study both the classical and plus/minus Selmer groups over the cyclotomic
$\mathbb{Z}_p$-extension of $F$. In particular, we give sufficient conditions
for these Selmer groups to not contain a non-trivial sub-module of finite
index. Furthermore, when $p$ splits completely in $F$, we calculate the Euler
characteristics of the plus/minus Selmer groups over the compositum of all
$\mathbb{Z}_p$-extensions of $F$ when they are defined.
