I have read in the paper of Meigniez "Submersions, fibrations and bundles" that a smooth surjective submersion $f: E \rightarrow B$ whose fibers are all diffeomorphic to $\mathbb{R}^{n}$ is locally trivial, i.e. a fiber bundle (corollay 31).

1) Is there a counterexample for $f$ not a submersion, which is not a fiber bundle even topologically?

2) Is there a smooth family of vector spaces all isomorphic to $\mathbb{R}^{n}$ which is not a vector bundle, even topologically?

3) Is there a surjective smooth map $f: E \rightarrow B$, with $E$ and $B$ compact, such that all the fibers are pairwise diffeomorphic, but which is not a fiber bundle, even topologically? (Here of course I do not require any more that the fibers are diffeomorphic to $\mathbb{R}^{n}$)