By Mordell-Weil, for any number field $K$ we have

$$C(K)=\mathbb{Z}^r \times E(K)_{\mathrm{tors}}$$

As you mention, Mazur showed all the possible options for $E(\mathbb{Q})_{\mathrm{tors}}$ in his famous 1977 paper.

The only other $K$ for which we have a torsion theorem are the quadratic fields. This is the result of a long series of papers by Kamienny, Kenku and Momose from 1982 to 1992. In particular,

$$\begin{equation}
E(K)_{\mathrm{tors}} \cong
\begin{cases}
\mathbb{Z}/m\mathbb{Z} & \text{for} 1\leq m\leq 18, m\neq 17 \\
\mathbb{Z}/2\mathbb{Z} \oplus\mathbb{Z}/2m\mathbb{Z} & \text{for}\,\, 1\leq m\leq 6 \\
\mathbb{Z}/3\mathbb{Z} \oplus\mathbb{Z}/3m\mathbb{Z} & \text{for}\,\, 1\leq m\leq 2 \\
\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}
\end{cases}
\end{equation}$$

For number fields of degree $>2$ the problem is open, although quite a lot is known for degree $\leq 5$. A (probably not up-to-date) survey on partial results:

- Andrew Sutherland, "Torsion subgroups of elliptic curves over number fields" (2012)

For $K$ a local field, I think it is an old result of Mattuck that $C(K)$ is a topologically finitely generated abelian profinite group. Not sure if anything else is known.