A procedure is developed and applied to characterize the fluctuations in global shape and folding features, in simple molecular chains, as a function of the temperature (T). The approach uses the novel concept of probability of the number of overcrossings associated with all rigid placements of a given configuration as a tool to describe the compactness of a chain, as well as the topology and distribution of its entanglements. A number of shape descriptors are derived from this notion. One of the descriptors provides an absolute characterization of molecular shape, using a value of 1 for stretched, linear chains and a value of 0 for compact and entangled ones, regardless of their size and anisometry. The averages of these descriptors along molecular dynamics trajectories (or over configurational space) can be studied as a function of T. Fluctuations in these averages serve to assess the flexibility of a molecular configuration regarding its deformation into a new fold. The procedure is applied to two models of simulated "melting" of an initially linear, all-trans configuration of dodecane. A broad transition is recognized in terms of the shape descriptors, from the stretched chains at low T to the folded chains at high T. The results provide a quantitative expression of the view that melting of some crystalline, linear polymer chains can be interpreted, in a first approximation, in terms of the molecular rigidity and the dominant types of entanglements. A similar transition is also found when studying the shape descriptors as a function of excluded volume in the simulation of polymer swelling with a simple necklace model. Our results suggest a new viewpoint to relate configurational transitions with fluctuations in the macromolecular shape.