Let $\Sigma$ be an oriented closed surface and $E$ be the direct sum of $T\Sigma$ with a trivial line bundle. Is $E$ a trivial rank $3$ vector bundle?
For genus $0$ and $1$ the answer is yes since one easily finds trivializations. For genus $\geq 2$, I can't find a trivialization, but $E$ has no non-vanishing characteristic class either to disprove the trivialness.
You may embed your oriented closed surface into $\mathbb R^3$ in the usual way.
Then the normal bundle is trivial, and of course the tangent bundle of $\mathbb R^3$ is trivial, so that the same is true of its restriction to $\Sigma$.
Thus we find that indeed, $$T\Sigma \oplus \text{ rank one trivial (i.e. normal bundle to $\Sigma$ in $\mathbb R^3$) }$$ $$ = \text{ rank three trivial (i.e. restriction to $\Sigma$ of $T\mathbb R^3$) }.$$