Fundamental group of R^2 minus the (ir)rationals Let
$$E =\{(x,0) \in \mathbb{R}^2 \colon x \in \mathbb{Q} \}$$
$$F = \{(x,0) \in \mathbb{R}^2 \colon x \in \mathbb{R} \setminus \mathbb{Q}\}$$
compute the fundamental group of $\mathbb R^2\setminus E$ and $\mathbb R^2\setminus F$.
How can I start?
(I don't know why the { symbols don't appear)
 A: When you want to compute the fundamental group of a wild space very often the thing to do is identify it as the subgroup of an inverse limit of simpler fundamental groups (often the first shape group). A result of Fischer and Zastrow says that if $X\subseteq \mathbb{R}^{2}$, then the canonical homomorphism of $\pi_1(X,x)$ into the shape group $\check{\pi}_{1}(X,x)$ is injective  for any $x\in X$. Of course, this homomorphism is not always injective (even for 2-dimensional compacta) but for a subset of the plane like you have this approach should work. This is, for instance, how you compute the fundamental group of the Hawaiian earring. I would begin by looking for some simple approximating spaces (probably with free fundamental groups) with projection maps and figuring out which elements of the inverse limit of the fundamental groups of these spaces are represented by loops.
Here is the paper I mentiond:
Fischer, Zastrow, The fundamental groups of subsets of closed surfaces inject into their first shape groups. Algebraic and Geometric Topology. Volume 5 (2005) 1655–1676. 
A: These groups are rather ugly.
I don't know what one might possibly mean by "computing" them.
Here's an example of a path that you could use to construct an element in $\pi_1(\mathbb R^2\setminus E)$: the graph of the function $y=x\sin(x)$ (appropriately shifted so that it doesn't cross the x-axis at the origin).
You can construct elements of $\pi_1(\mathbb R^2\setminus F)$ with similarly pathological behaviour.
A: One point to start is to look at recent work by R. Diestel and P. Sprüssel
The fundamental group of a locally finite graph with ends, to appear in Advances in Mathematics
Your examples are probably not covered by their results, but maybe you can use the methods.
