  Let $E$ be an elliptic curve over $\mathbb{Q}$ with supersingular reduction
at $p$ with $a_p=0$. We study the asymptotic growth of the plus and minus
Tate-Shafarevich groups defined by Lei along the cyclotomic
$\mathbb{Z}_p$-extension of $\mathbb{Q}$. In this paper, we work in the general
framework of supersingular abelian varieties defined over $\mathbb{Q}$. Using
Coleman maps constructed by Buyukboduk--Lei, we define the multi-signed
Mordell-Weil groups for supersingular abelian varieties, provide an explicit
structure of the dual of these groups as an Iwasawa module and prove a control
theorem. Furthermore, we define the multi-signed Tate-Shafarevich groups and,
by computing their Kobayashi rank, we provide an asymptotic growth formula
along the cyclotomic tower of $\mathbb{Q}$.
