Are viscoelastic materials to be considered solids or fluids?
There are two basic models that describe the two fundamental branches of viscoelastic behaviour:


The Maxwell model is made up of a spring and a dashpot in series: For fast deformations, the dashpot does not really move, and your spring gives pure elastic behaviour. For slow deformations, the spring stays relaxed, and the damper fluid in your dashpot turns all energy into heat. 
The Voigt-Kelvin/Kelvin-Voigt model is a spring and dashpot in parallel: Even at the slowest/smallest deformation, the spring never relaxes until you let go and let it return your sample to the original shape. And the faster you try to deform it (e.g. oscillatory), the more heat your damper produces.


The former, e.g. a molten polymer, is a liquid, the latter, e.g. crosslinked rubber, is a solid.