(Not quasicoherent)
On \(X = \Spec A\) with \(Z \subseteq X\) closed and \(j: U = X \setminus Z \injects X\) open, the sheaf \(j_!\OO_U\) extended by zero is an \(\OO_X\)-module that is not quasicoherent: its sections are not a localization at any point of \(Z\). The constant sheaf \(\ul{\ZZ}\) on a positive-dimensional \(X\) is another, for the same reason — restriction maps do not localize.
(Quasicoherent, not coherent)
\(\OO_{\AA^1}\) pushed to a point: \(f: \AA^1_k \to \Spec k\) has \(f_*\OO_{\AA^1} = \widetilde{k[t]}\), quasicoherent but not finitely generated over \(k\). On any integral \(X\), the constant sheaf of rational functions \(\mck_X\) with \(U \mapsto K(X)\) is quasicoherent and not coherent.
(Coherent, not locally free)
On \(X = \Spec k[t]\), the skyscraper \(\widetilde{k[t]/(t)}\) at the origin is coherent, with fibre dimension \(1\) at the origin and \(0\) elsewhere. Any torsion module gives such an example, and the non-constant fibre dimension is exactly the obstruction. On a singular curve the maximal ideal sheaf at the singular point is coherent, torsion-free, and still not locally free.
The three examples show that each implication in the hierarchy is strict. They separate the failure mechanisms: quasicoherence can fail through incompatibility with localization, coherence through failure of finite generation, and local freeness through varying fibre dimension. Over a reduced Noetherian scheme, a coherent sheaf is locally free exactly when \(x \mapsto \dim_{\kappa(x)} \mcf \tensor \kappa(x)\) is locally constant.