In this work damping properties of bending viscoelastic thin structures are enhanced by topology optimization. Homogeneous linear viscoelastic plates are optimized and compared when modeled by either the Kirchhoff-Love or Reissner-Mindlin plate theories as well as by the bulk 3D viscoelastic constitutive equations. Mechanical equations are numerically solved by the finite element method and designs are represented by the level-set approach. High performance computing techniques allow to solve the transient viscoelastic problem for very thin 3D meshes, enabling a wider range of applications. The considered isotropic material is characterized by a generalized Maxwell model accounting for the viscoelasticity of both Young modulus and Poisson's ratio. Numerical results show considerable design differences according to the chosen mechanical model, and highlights a counterintuitive section shrinking phenomenon discussed at length. The final numerical example extends the problem to an actual shoe sole application, performing its damping optimization in an industrial context.