Additive Subgroups of the Reals. Does anyone know if there is a classification of the subgroups of the real numbers taken under addition? If not can anyone point me in the directiong of any papers/materials which discuss properties of or interesting facts about these subgroups? 
 A: The 2002 paper of Simon Thomas in JAMS provides a precise measure of "hopelessly complicated". As the rank of the groups increases, so does the complexity of the classification problem.
A: For every real number $\alpha$ with $0<\alpha<1$, there is an uncountable subgroup of $\mathbb{R}$ with Hausdorff dimension $\alpha$: see e.g. https://perswww.kuleuven.be/~u0018768/artikels/actions-free-group.pdf
A: If you're interested in topological classification, then this might be useful: Farah and Solecki - Borel subgroups of Polish groups, Advances in Mathematics 199, 2006, 499-541.
Among a lot of other things, one of their results shows that for any countable ordinals $\alpha \neq 2$ and $\beta \geq 2$, there are $\Pi_{\alpha}^{0}$-complete and $\Sigma_{\beta}^{0}$-complete additive subgroups of any uncountable polish group. For connected abelian polish groups, this was previously shown by Mauldin by refining a result of Klee.
A: Slightly off-topic, a weird subgroup in two dimensions is constructed in this paper: 
Ryuji Maehara. On a connected dense proper subgroup of ${\bf R}^2$ whose complement is connected . Proc. Amer. Math. Soc. 97 (1986) 556-558. MR 840645. 
That subgroup is constructed as a graph of a wild group homomorphism $f:\mathbf{R}\to \mathbf{R}$, so it is also a subgroup:
$\{ (x,f(x))  \mid x\in \mathbf{R}\}$
A: Every torsion-free abelian group of cardinality at most $2^\omega$ is isomorphic to a subgroup of the reals. (To see this, note that any such group can be embedded in a divisible torsion-free group of the same cardinality, i.e., a vector space over $\mathbb Q$, which can in turn be embedded in any other vector space over $\mathbb Q$ of the same or greater dimension.) Since already the structure of rank 2 abelian groups is hopelessly complicated, you are not going to find any sensible classification.
A: If you would like to classify the subgroups in the sense of Lebesugue measure, you may find the following facts helpful.
(1) Any measurable  proper subgroup of the real line is of measure $0$.
(2) Any non-measurable subgroup $G$ of the real line charges fully everywhere, i.e., for any interval $I$, $m^{\ast}(G \cap I)=|I|$, where $m^{\ast}(\cdot)$ denotes the outer Lebesgue measure.
(3) Non-measurable subgroup of the real line exists.
A: I'm surprised no one has yet stated the most obvious fact (though I guess Xianghong's answer comes pretty close), which is that an additive subgroup of the reals is either of the form $a\mathbb{Z}$ or is dense in the real line (an obvious consequence from division with remainder).
A: You may be interested in the following elementary proofs:
https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxzb251bWF0aHMyfGd4OjhhZTM3MmVkMWJiN2UzMA
