Possible groups of K-rational points for elliptic curves over arbitrary fields

The structure of $E(K)$ for $K$ a complete local field, say a finite extension of $\mathbb Q_p$ or $\mathbb C_p$, is quite standard. Let $E_0(K)$ denote the set of points with good reduction. Then there are exact sequences $$ 0\to E_0(K)\to E(K) \to \Phi \to 0 $$ and $$ 0 \to E_1(K) \to E_0(K) \to \tilde E^{\text{ns}}_p(\mathbb F) \to 0 .$$ Here $\Phi$ is a finite group, the group of components on the Neron model. It is either cyclic of order $\text{ord}_v(\text{Disc}(E/K))$, or it is one of the groups $1,C_2,C_3,C_4,C_2\times C_2$.

$\tilde E^{\text{ns}}_p(\mathbb F)$ is the group of non-singular points defined over the residue field $\mathbb F$. In the case of singular reduction, it is either the additive group $\mathbb F^+$, or a (possibly) twisted form of the multiplicative group $\mathbb F^*$.

Finally, $E_1(K)$ is isomorphic to the group of points of a formal group. In particular, if $K$ is a finite extension of $\mathbb Q_p$, it has a subgroup of finite index that isomorphic to the additive group of the ring of integers of $K$.

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.

The answer to one possible interpretation of the title question -- vary over all elliptic curves over all fields and ask which groups arise -- is given in this paper.

With regard to the structure of Mordell-Weil groups of elliptic curves over local fields, I think you will find that $\S$5.1 of this joint paper with Allan Lacy relevant. (The title is "Mordell-Weil Groups of Abelian Varieties Over Local Fields".) The case of characteristic $0$ is a slightly more general and less precise variant of Joe Silverman's answer, but things work a bit differently in positive characteristic.