The set of admissible configurations and diffusive motions of chain molecules (beads and rigid links) on the quadratic lattice are considered with respect to a set of ``flip rules''; these permit configurations to change and the molecules to diffuse by flipping one bead at a time consistent with the excluded volume requirement. The set of configurations splits up into families such that configurations within a family can flip into each other but no transitions between configurations in different families are possible under the flip rules. The families are designated as mobile or sessile, the former including those that are able to diffuse to arbitrarily large distances; the sessiles must remain within a restricted region under the flip rules no matter how many flips are permitted. A complete discussion of chain configurations on a channel one link wide is given and the general situation for wider channels and the infinite plane is discussed. The importance of such considerations for entropy calculations and their possible relevance for biological phenomena are indicated.