The energy landscape of two-dimensional lattice polymers is investigated, using the so-called lid algorithm. We find that a multitude of pockets around local energy minima exist that exhibit approximately exponentially growing local densities of states. This behavior is similar to the energy landscape of e.g. two- and three-dimensional covalent lattice networks and the energy landscape of spin glasses. However, in contrast to these systems, we find that two classes of polymer landscapes exist, depending on the length of the polymers. As a consequence, short polymers appear to show a much more "liquidlike" behavior, while long polymers resemble much more closely the glassy covalent networks.