Suppose that $M$ is a smooth manifold. Over each coordinate chart $(U,\varphi)$ $M$ looks like euclidean space. In particular the tangent bundle $TM$ is trivial over $M$. So you the covering of $M$ by "trivialisations" of $TM$ is at most as rich as the covering of $M$ by charts. One can argue similarly for any vector bundle over $M$. However it may happen that $M$ is complicated but has trivial tangent bundle so the covering of $M$ by trivialisation consists from only one element.
Is it true that for each (say closed) manifold $M$ one can find a vector bundle $E \to M$ such that each trivialization $U$ of $E$ the manifold $M$ is euclidean over $U$?