Let $M$ be a smooth $m$-manifold and $N \subseteq M$ a smooth $n$-dimensional submanifold. Then there is a so called tubular neighborhood $(E,p)$ of $N$ in $M$, which is a neighborhood $E$ of $N$ in $M$ together with a vector bundle $$ p \colon E \to N $$ of rank $m-n$ whose zero section is $N$.
Is the vector bundle $p$ always orientable or is there always a tubular neighborhood $(E,p)$ for which $p$ is orientable? This would be useful to apply the Thom isomorphism theorem.
Consider $M$ the (open) Moebius band and $N\subset M$ the central circle in $M$. Then the total space of the normal bundle of $N$ is diffeomorphic to $M$ and, hence, is nonorientable. Same for the tubular neighborhoods of $N$ in $M$.