For ethylene copolymers, secondary relaxations control many important functional properties, such as room temperature ductility and subambient impact toughness. The location and intensity of these relaxations could also indicate the local environments of the polymer main chain. For example, the "crankshaft" motions for the methylene sequences give rise to the gamma transition and the beta relaxation introduced by the side branches. We have undertaken a broad based study of several families of copolymers of widely different chemical compositions including vinyl acetate, methyl acrylate, acrylic acid, and alpha-olefins. Comparisons have been obtained both from within the family as a function of co-monomer content and between families for the effect of the chemically different side groups. Interesting contrasts have been found. For example, the relative intensity of the beta relaxation versus that of the alpha transition is a function of the crystallinity. Within the family of the ultralow density polyethylenes derived from different polymerization processes, the breadth of the beta relaxation is found to be strongly related to the homogeneity of the short chain distribution as independently measured by the temperature rising elution fractionation (TREF) technique.