There are many examples of unstable bundles on a projective surface that have no non-trivial subbundles. 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 $$0 \longrightarrow \mathcal O \longrightarrow E \longrightarrow I(k) \longrightarrow 0$$ on $\mathbb P^2$ in which $E$ is locally free. Furthermore, the Chern classes of $E$ are $c_1(E) = k$ and $c_2(E) = m$ (for this, see page 103 of "Vector bundles on complex projective spaces", by Okonek, Schneider and Spindler). If $k < -3$, 0$, this vector bundle is clearly unstable; but for most values of$k$and$m$it can not split as a direct sum of line bundles, hence it cannot contain a line subbundle (since every extension of line bundles on$\mathbb P^2$splits). 3 deleted 29 characters in body There are many examples of non-semistable unstable bundles$E$on a projective surface which are unstable, but that have no non-trivial subbundles. For example, if$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 $$0 \longrightarrow \mathcal O \longrightarrow E \longrightarrow I(k) \longrightarrow 0$$ on$\mathbb P^2$in which$E$is locally free. Furthermore, the Chern classes of$E$are$c_1(E) = k$and$c_2(E) = m$(for this, see page 103 of "Vector bundles on complex projective spaces", by Okonek, Schneider and Spindler). If$k < -3$, this vector bundle is clearly unstable; but for most values of$k$and$m$it can not split as a direct sum of line bundles, hence it cannot contain a line subbundle (since every extension of line bundles on$\mathbb P^2$splits). 2 added 452 characters in body; added 1 characters in body There are many examples of non-semistable bundles$E$on a projective surface which are not unstable, but have no non-trivial subbundlesubbundles. 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 $$0 \longrightarrow \mathcal O \longrightarrow E \longrightarrow I(k) \longrightarrow 0$$ on$\mathbb P^2$in which$E$is locally free. Furthermore, the Chern classes of$E$are$c_1(E) = k$and$c_2(E) = m$(for this, see page 103 of "Vector bundles on complex projective spaces", by Okonek, Schneider and Spindler). If$k < -3$, this vector bundle is clearly unstable; but for most values of$k$and$m$it can not split as a direct sum of line bundles, hence it cannot contain a line subbundle (since every extension of line bundles on$\mathbb P^2\$ splits).