I hope this question is reasonable enough to have a well known answer. i.e either there is a simple invariant (like the homotopy groups) that characterizes the homeomorphism type of such set among other such sets or there is an explanation of why such classification is difficult. partial results, like the well known fact that every two open connected and simply connected subsets of the plane are homeomorphic, would be interesting too. thanks.
[Math] classification of open subsets of euclidean space up to homeomorphism
at.algebraic-topologyclassificationgn.general-topologygt.geometric-topology
Best Answer
If you have a presentation $P$ of a group $G$ with finitely many generators and relations then you can construct a $2$-dimensional simplicial complex $K(P)$ with $\pi_1(K(P))=G$. You can then embed $K(P)$ in a Euclidean space (I think $\mathbb{R}^5$ is good enough) and take a regular neighbourhood $U$. Then $U$ is an open subset of $\mathbb{R}^5$ that is homotopy equivalent to $K(P)$ with $\pi_1(U)=G$. The theory of finite group presentations is undecidable, so this means that there is no effective classification of such subsets. This is without any kind of local wildness. You can construct much worse examples by taking the complements of wild knots, fractals, the Alexander horned sphere and so on.