Is there a classification of all countable subgroups of the circle $\mathbb{T} \simeq \mathbb{R}/\mathbb{Z}$?

It seems that there are quite a lot of them, e.g.:

- cyclic subgroups $\{a^n\colon n\in\mathbb{Z}\}$
- finite subgroups
- subgroups of the form $\{k/2^n\colon k,n\in \mathbb{Z}\}$ or something similar
- direct sums of the above...

Equivalently, is there a classification of all compact monothetic groups? (Each countable subgroup of the circle can be realized as a group of eigenvalues of some probability-preserving transformation and vice versa).