Suppose $M$ is a compact, smooth $n$-manifold, $X$ is a smooth vector field on $M$, and $\omega$ is a smooth $n$-form on $M$. Then is it true that the Lie derivative $L_X \omega$ is not nowhere vanishing, i.e., $(L_X \omega)|_p=0$ for some $p\in M$?
I know that an exact $1$-form on a compact manifold is not nowhere vanishing, so I tried to prove similarly, but I couldn't.
If $M$ is orientable without boundary and $\omega$ is nowhere vanishing, then this would be true, because by Cartan's formula we have $L_X \omega =d \iota_X \omega$, so by Stokes' theorem it follows that $\int_M L_X\omega=\int_M d\iota _X\omega=\int_{\partial M} \iota_X \omega=0$ because $M$ is boundaryless.
You essentially solved this for the case that $M$ is orientable. Let's see the non-orientable case. By Cartan's formula we also have that
$$ L_X\omega \;\; =\;\; d(i_X\omega) \;\; =\;\; \eta $$
where $\eta$ is some other top-dimensional form. However, given that $M$ is non-orientable then $M$ does not possess a non-vanishing top form. Therefore $\eta_q = 0$ for some $q \in M$.