For a vector bundle $E$, I will denote the maximum number of linearly independent global sections of $E$ by $\eta(E)$. We have $\eta(E) \in \{0, 1, \dots, \operatorname{rank}(E)\}$ and $\eta(E) = \operatorname{rank}(E)$ if and only if $E$ is trivial - for this reason, $\eta(E)$ should be a measure of how close a given bundle is to being trivial.
There are many examples where $\eta(E) = \operatorname{rank}(E)$, for example $E = T\mathbb{R}^n$. By a fairly well known result, $\eta(TS^n) = n$ if and only if $n = 0, 1, 3,$ or $7$; more generally, $\eta(TG) = \dim G$ for any Lie group $G$. For the other values of $n$ we can say more due to the Hairy Ball Theorem. Namely, if $n$ is even, $\eta(TS^n) = 0$, and if $n$ is odd, $\eta(TS^n) \geq 1$.
Is $\eta(E)$ a useful piece of information; in particular, does it have a name? Are there any methods to calculate $\eta(E)$?
Michael, read about characteristic classes. For example, the Poincaré dual of $c_k(E)$ (the $k$th Chern class of a complex bundle of rank $n$) is the cycle along which $n-k+1$ generic sections become linearly dependent. One can also deduce directly from the behavior of Chern classes and short exact sequences of bundles, if there is a trivial subbundle of rank $k$, all Chern classes $c_j(E)=0$ for $j>n-k$. There are analogous statements for Stiefel-Whitney and Pontryagin classes of real bundles.