Consider $\mathbb{Z}/p^k\mathbb{Z}$ as $\mathbb{Z}$-module, where $p\in \mathbb{Z}$ is a prime. What is, up to isomorphism, the localization $(\mathbb{Z}/p^k\mathbb{Z})_{(p)}$ as a $\mathbb{Z}$-module?
It seems to me that $(\mathbb{Z}/p^k\mathbb{Z})_{(p)}$ must be something like $$(\mathbb{Z}/p_1^{r_1}\mathbb{Z}) \oplus(\mathbb{Z}/p_2^{r_2}\mathbb{Z})\oplus \dots \oplus (\mathbb{Z}/p_n^{r_n}\mathbb{Z}),$$ because it is a finitely generated $\mathbb{Z}$-module and it has a non trivial torsion. And here comes my problem: We know that $S^{-1}A$ as $A$-modules are flat but we know also that $\mathbb{Z}/m\mathbb{Z}$ is not flat as a $\mathbb{Z}$-module. Where have I made a mistake?
Note that $\Bbb{Z}/p^k\Bbb{Z}$ is a local ring with maximal ideal $(p)$, so the map $$(\Bbb{Z}/p^k\Bbb{Z})_{(p)}\ \longrightarrow\ \Bbb{Z}/p^k\Bbb{Z}:\ \tfrac{a}{b}\ \longmapsto\ ab^{-1},$$ is well defined, and in fact it is an isomorphism.
There is no mistake in your reasoning on flatness; indeed $(\Bbb{Z}/p^k\Bbb{Z})_{(p)}$ is flat as a $\Bbb{Z}/p^k\Bbb{Z}$-module because it is a localization thereof, but it is not flat as a $\Bbb{Z}$-module because it has $\Bbb{Z}$-torsion.