Energy transfer in polymers is characterized by nonexponential decays, mostly due to the disorder of the underlying medium. In this work we take as models for the medium small-world networks (SWNs). SWNs are built starting from regular lattices (say from linear chains) through the insertion (with probability p) of additional links, which then connect distant pairs of sites. In this way SWNs combine random and regular features. As a dynamical problem we evaluate the energy migration followed by trappin g (quenching) by acceptors, randomly distributed over the SWN, and compare the trapping decay to the forms found when the underlying structures are regular lattices, fractals or ultrametric spaces; as we show, trapping on SWNs displays new decay aspects.