Assume $p$ is prime. By $\mu_p$ I mean of course the $p$-th roots of unity.
More specifically I'm working in the context of the $p$ torsion of elliptic curves.
I guess having prime order narrows it down a lot with respect to what these modules can be, but I can't think of a proof for this statement.
No, this is far from true. You are asking: how many maps are there from the absolute Galois group of (whatever your field is) to $\mathbb{F}_p^\times$.
There are a lot (unless your field is algebraically closed or something). If nothing else, you have $\mu_p^{\otimes n}$ for any $n$.