Instability, fear of falling and fall injuries are common problems for the older population and individuals with chronic disabilities. Development of screening tools and outcome measures of balance performance and fall risk has become of great interest for health professionals and researchers in the field of rehabilitation. Although linear modeling is desirable due to its simplicity, however, most physiological systems are too complicated to lend themselves to linear analysis because the human body is multi-segmental, and there are feedforward and feedback control schemes of mutual relationship between the segments. In this study, we investigated the use of nonlinear dynamic tools to extract characteristic features of postural sway. Among various methods of nonlinear dynamics, the Rényi fractal dimension and the Rényi spectrum are presented as quantitative descriptors. The center of foot pressure (COP) was recorded during four quiet standing tasks of increasing difficulty (eliminating vision and use of a compliant surface). The COP trajectories' Rényi dimension and Rényi spectra of young healthy subjects with no history of neurological disorder were investigated in comparison to that of elderly patients with balance impairment and history of frequent falls. Sway path, the common linear parameter of postural sway, was also used to investigate its distinction from non-linear parameters. The results of this study suggest the COP trajectories' Rényi dimension and sway path can be used as classifiers for balance disorders, and they provided different indications of postural control system characteristics between the two groups and different task demands.