A short exact sequence of vector bundles has a canonical splitting

Consider the split exact sequence of trivial line bundles over any nonempty manifold $M$, $0\to M\times \mathbb{R}\to M\times {\mathbb{R}}^2\to M\times \mathbb{R}\to 0$, where the first map is $t\mapsto (t,0)$ and the second is projection to the second coordinate. The map $s_0(p,u)=(p,(0,u))$ is an explicit smooth splitting.

For every $c\in \mathbb{R}$, the bundle automorphism $T_c(p,(x,y))=(p,(x+cy,y))$ fixes the included first summand and commutes with projection to the second summand, so it is an automorphism of the exact sequence in step 1.1. Every splitting has the form $s_a(p,u)=(p,(a(p)u,u))$ for a smooth function $a$, while $T_c\circ s_a=s_{a+c}$. Thus no splitting is fixed by all automorphisms of the sequence: taking $c=1$ moves every candidate. A splitting determined canonically by the sequence would have to be invariant under these automorphisms, so none exists.