Local splitting when a quotient sheaf is locally free

In a short exact sequence of sheaves $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$, if $\mathcal{H}$ is locally free of finite rank, the sequence splits near every point. Trivialize $\mathcal{H}$, lift its finitely many basis sections locally, and shrink to one neighbourhood on which all lifts exist. The lifts define a splitting. If $\mathcal{F}$ is also locally free, then so is $\mathcal{G}$. Conversely locally free kernel and middle term need not give a locally free quotient: multiplication by $t$ on ${\mathcal{O}}_{\mathrm{Spec}k[t]}$ has quotient supported at the origin.