One important case when vanishing of the Euler class does imply triviality of the bundle (and hence existence of nowhere zero section) is for oriented rank 2 bundles over paracompact bases. In fact, Euler class gives one-to-one correspondence between the set of isomorphism classes of such bundles and the second cohomology group [Husemoller's "Fiber bundles" book, 20.2.6].

If rank is $>2$, then a vector bundle is determined by Euler and Pontryagin classes up to finite ambiguity (provided the base is a finite cell complex). In rare cases Euler and Pontryagin determine the bundle completely.

For example, rank 4 bundles over $S^4$ are in one-to-one correspondence with $\pi_3(SO(4))\cong\mathbb Z+\mathbb Z$ where the latter isomorphism can be chosen so that the bundle $(n,m)$ has Euler class $n$ and first Pontryagin class $2m$. If memory serves me, this can be found in Milnor's original paper on exotic 7-sphere, but there are more recent detailed sources, e.g. see here.