$U(1)$ bundles over a flat Euclidean $\mathbb{R}^d$ or Minkowski $\mathbb{R}^{d,1}$ space

Question

I know that we can have a $U(1)$ bundle over the base manifold like $S^2$. For example, we can choose the fibre $U(1)=S^1$ over $S^2$ as the $S^3$ Hopf fibration.

Now, can we have any nontrivial $U(1)$ bundles over a flat Euclidean $\mathbb{R}^d$ or Minkowski $\mathbb{R}^{d,1}$ space?

If so, could you give some examples?

If not, could you give me a proof? (or at least sketch your arguments to convince me?)