= Möbius line bundle
{c}
{title2=$L\to S^1$}
The Möbius line bundle over the circle is
$$
L=(\mathbb R\times\mathbb R)/((t+2\pi,a)\sim(t,-a)),
$$
with projection to $t$ modulo $2\pi$. It is a smooth rank-one <vector bundle>. A section is represented by a function $s$ satisfying $s(t+2\pi)=-s(t)$, so the <intermediate value theorem> forces every <continuous> section to vanish somewhere. Thus it has no <nowhere-zero section> and is not isomorphic to the <trivial vector bundle> of rank one.
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