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There are also examples that are characterized dynamically. By Viterbo, every compact, contact-type hypersurface of $\mathbb{R}^{2n}$ carries a closed Reeb orbit, i.e. a closed leaf of the characteristic line bundle. (Actually, Hofer and Zehnder show this is true for any stable hypersurface.) There are examples of autonomous Hamiltonians on $\mathbb{R}^{2n}$ that fail to have any closed orbits on a specific level set, thus providing examples of hypersurfaces that fail to be stable. These are due to Ginzburg and Gurel.