Möbius line bundle ($L\to S^1$)

The Möbius line bundle over the circle iswith 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.