Suppose we have a manifold $M$ of dimension $m$ and let $E$ be the rank $n$ trivial vector bundle on $M$. Let $\Gamma$ be a section of the trivial bundle $E$. My question is, are there some conditions on the section $\Gamma$ which ensure that it extends to a basis of sections of the vector bundle $E$.
I know that in general this is not possible (for example in case of the trivial bundle on sphere) but I am unable to see what exactly is the obstruction.
The trivial bundle defines a line subbundle $L \subset E$; you're asking precisely that $E/L$ also be trivial. But as you point out in the question, the tangent bundle $TS^2$ plus a trivial line bundle is the trivial 3-plane bundle, so there is a line subbundle of the trivial 3-plane bundle that gives $TS^2$ as its quotient (and hence every other section of the trivial rank 3 bundle must, at some point, be a scalar multiple of our "special" one.)