Let $n$ be odd. Recall that $S^n$ is parallelisable if and only if $n = 1, 3, 7$. For every other $n$, there exists $k < n$ such that $S^n$ admits $k$ linearly independent vector fields, but not $k + 1$. As $n$ is odd, $\chi(S^n) = 0$ so $S^n$ admits a nowhere-zero vector field by Poincaré-Hopf; that is, $k > 0$. Therefore $TS^n = E\oplus\varepsilon^k$ where $E$ has rank $n - k$ and does not admit a nowhere-zero section. Note however that $e(E) \in H^{n-k}(S^n; \mathbb{Z}) = \{0\}$ as $0 < n-k < n$.

It should be noted that the precise value of $k$ is known for every $n$ by work of Radon, Hurwitz, and Adams. Namely $k = \rho(n+1) - 1$ where $\rho(n+1)$ denotes the $(n+1)^{\text{st}}$ Radon-Hurwitz number: if $n + 1 = 2^{4a+b}c$ where $a \geq 0$, $0 \leq b \leq 3$, and $c$ is odd, then $\rho(n+1) = 8a + 2^b$.