I'm pretty sure the answer is out there, see this MathOverflow question, but that is unfortunately way over my head :).
I'm interested in the case that $G$ is a Lie group (e.g. $U(1)$), but I don't need to know the answer for general manifolds -- just for boring old Euclidean space.
For example, I sort of feel like there is only one $U(1)$-bundle over $\mathbb{R}^n$ for any $n$ (up to isomorphism), and possibly only one $SU(2) \times U(1) \times SU(3)$-bundle, too. But I have no idea how to prove/disprove that. Do I need to go ahead and try to understand these classifying spaces referenced in the link above? Or is there a more elementary argument?
If there can be multiple different $G$-bundles over $\mathbb{R}^n$, what is the simplest Lie group $G$ and the lowest value of $n$ for which this occurs? Is there a simple explicit construction?
The $G$-bundles over $R^n$ are trivial (i.e they are isomorphic to $R^n\times G$) for any lie group $G$ since $R^n$ is contractible.