I am wondering if there are necessary and sufficient conditions under which an one-dimensional subbundle of $TM$ has a nowhere vanishing vector field.

More precisely let $M$ be a compact smooth manifold.

a. When dose there exist a one-dimensional (smooth or continuous) subbundle $L\subset TM$?

b. If $L\subset TM$ is a continuous/smooth line subbundle of $TM$, does there exist a nowhere vanishing continuous/smooth section $X:M\to L$? If so, the euler characteristic of $L$ should be zero.

This is related to the partially hyperbolic system $f:M\to M$ and $TM=E^s\oplus E^c\oplus E^u$. I am curious if there is a 'center flow' if $\dim E^c=1$.

To Ryan: Am I right to say the following about your answers:

If there exists an 1-dimensional subbundle $L$ of $TM$, then $\chi(M)=0$. This is independent of the case whether $M$ is orientiable or not.

If $M$ is orientable, then there always exists an orientable 1-dimensional subbundle $L$ of $TM$.

Another question is, when is an 1-dimensional subbundle $L\subset TM$ orientable? Is it sufficient to assume that $M$ is orientable?

Thank you all. I did not formulate some questions properly. What I really mean is:

I. For a given line bundle $L\subset TM$, what is the obstruction for $L$ beging orientable? (or equivalently trivial according to Georges)

For example let $L_{\mathbb{C}}$ be a complex line bundle over a complex manifold $M$, if the top Chern class $c_1(L_{\mathbb{C}})$ does not vanish, then $L_{\mathbb{C}}$ can not be trivial. Is there some similar results in the real case?

II. Is there an example such that $M$ is orientable and has a non-orientable line bundle $L\subset TM$?