There are many examples of non-semistable bundles $E$ on a projective surface which are not unstable, but have no non-trivial subbundle. For example, if $k$ is an integer with $k < 3$ and $I$ is the sheaf of ideal of $m$ distinct points in $\mathbb P^2$, with $m > 0$, there exists an extension
$$
0 \longrightarrow \mathcal O \longrightarrow E \longrightarrow I(k) \longrightarrow 0
$$
in which $E$ is