  An infinite hierarchy of layering transitions exists for model polymers in
solution under poor solvent or low temperatures and near an attractive surface.
A flat histogram stochastic growth algorithm known as FlatPERM has been used on
a self- and surface interacting self-avoiding walk model for lengths up to 256.
The associated phases exist as stable equilibria for large though not infinite
length polymers and break the conjectured Surface Attached Globule phase into a
series of phases where a polymer exists in specified layer close to a surface.
We provide a scaling theory for these phases and the first-order transitions
between them.
