We study Rayleigh-Taylor instability driven by a time-varying acceleration with power-law time dependence in two and three dimensional flows by employing group theory approach. We identify the structure of fields in Rayleigh-Taylor unstable flow, and focus on the scale-dependent dynamics of bubbles and spikes constituting Rayleigh-Taylor coherent structures. We find the time-dependence of the interface growth and growth-rate in the linear regime and we employ regular asymptotic expansions to derive the nonlinear solutions, including the interface velocity and curvature as well as the interfacial shear function. There is a family of nonlinear solutions, which can be parameterized by the shear function and/or by the curvature of the bubble or spike. In this family, the bubble solutions are shown to be regular, whereas spike dynamics may be singular. We show that the nonlinear dynamics is multiscale, with both the amplitude and wavelength contributing, and interfacial, with intense fluid motion near the interface and effectively no motion away from the interface and with vortical structures appearing at the interface. By further examining the nonlinear dynamics of both three-dimensional and two-dimensional systems, we find a geometry independent form of the nonlinear dynamics. This geometry-independent form is further employed to examine the effects of dimensionality and symmetry on the scale-dependent dynamics of Rayleigh-Taylor instability. We conclude by comparing our theoretical results against recent numerical simulations, finding excellent agreement and revealing some key features of Rayleigh-Taylor dynamics on the basis of our group theory approach.