I don't have a precise reference for the cases of $n=2$ or $n=3$, though I suspect that they could be found in Moise's "Geometric Topology in Dimensions 2 and 3".
However, I do have a nice reference for the amazing theorem (due to Stallings) that $\mathbb{R}^n$ has a unique $C^{\infty}$ structure for $n \geq 5$. It is written up very nicely in Steve Ferry's Geometric Topology Notes. See Chapter 10 starting on page 56. What he proves is that $\mathbb{R}^n$ has a unique PL structure, but that is the key to the result. The rest of the notes are also wonderful.
This theorem is surprising for a number of reasons. For instance, it says that if $\Sigma$ is an exotic $n$-sphere for $n \geq 5$, then $\Sigma \setminus \{p\}$ is homeomorphic to $\mathbb{R}^n$ for any point $p \in \Sigma$. In other words, the "exoticness" is "concentrated at a point". Of course, this also follows from the usual proof of the high dimensional Poincare conjecture using the $h$-cobordism theorem, which constructs a homeomorphism between a homotopy sphere and the usual sphere which is differentiable except at one point (that one point giving trouble due to the "Alexander trick").
Another remark that should be made is that Freedman and Donaldson proved that $\mathbb{R}^4$ has uncountably many $C^{\infty}$ structures.