Torsion groups of elliptic curves over some infinite abelian extensions of $\mathbb{Q}$

We determine, for an elliptic curve $E/\mathbb{Q}$, all the possible torsion groups $E(K)_{tors}$, where $K$ is the compositum of all $\mathbb{Z}_{p}$-extensions of $\mathbb{Q}$. Furthermore, we prove that for an elliptic curve $E/\mathbb{Q}$ it holds that $E(\mathbb{Q}(\mu_{p^{\infty}}))_{tors} = E(\mathbb{Q}(\mu_{p}))_{tors}$, for all primes $p \geq 5$ and $E(\mathbb{Q}(\mu_{3^{\infty}}))_{tors} = E(\mathbb{Q}(\mu_{3^3}))_{tors}$, $E(\mathbb{Q}(\mu_{2^{\infty}}))_{tors} = E(\mathbb{Q}(\mu_{2^4}))_{tors}$.