Given a trivial rank-$n$ vector bundle $V$ over a manifold $M$, there are many different trivializations of $V$, i.e. many possible choices of nowhere-dependent sections $\phi_1, \ldots \phi_n$. My question is,
What notions of "equivalent" and "inequivalent" are there for trivializations of $V$?
I am specifically interested in the case where $V = TM$, that is, parallelizable manifolds. In this restricted setting my question could be rephrased as,
When some construction starts with the phrase "Fix a trivialization $TM \cong M \times \mathbb R^n$..." then in what situations do I need to worry about the choice of trivialization?
(That's a pretty open-ended question but I'm interested in hearing any example that comes to your mind. Thanks)