This thesis is devoted to study the dynamical instability of some relativistic collapsing self-gravitating structures with both Newtonian and post-Newtonian approximations in metric f(R) gravity. In this setting, we consider evolution of spherical, cylin- drical and restricted axial stellar systems ¯lled with expansion and expansion-free adiabatic/non-adiabatic matter con¯gurations. We analyze the role of adiabatic index (sti®ness parameter) in the instability constraints of these self-gravitating structures. We construct dynamical equation using contracted Bianchi identities of the e®ective dark sources as well as usual matter distribution. The perturbation approach is ap- plied on physical variables and then formulate modi¯ed versions of collapse equation which leads to instability constraints at both N and pN regimes. We ¯rst consider spherically and cylindrically symmetric spacetimes ¯lled with charged and uncharged expansion-free anisotropic matter distributions. It is found that the adiabatic index does not have any role in the expansion-free evolution within N and pN approximations. Rather, this range is governed by pressure anisotropy, ra- dial pro¯le of system energy density, f(R) model and electric charge (for charged distribution). We also explore the role of heat radiations in non-viscous charged spherical and cylindrical systems as well as shearing viscous uncharged axial relativis- tic interior. It is found that heat radiations try to decrease stability of the evolving systems, while viscosity tends to increase system stability. The electromagnetic ¯eld decreases instability regions for charged spherical systems while its opposite e®ects have been observed for cylindrical collapsing self-gravitating systems. We conclude that f(R) dark energy sources coming up from the well-known f(R) models a®ect xi xii the whole dynamical behavior of collapsing systems due to its repulsive nature. Finally, we study the factors involved in the energy density irregularities of rela- tivistic planar °uid distribution in the presence of Palatini f(R) corrections. For this purpose, we develop a link between the Weyl scalar and structural properties of the system by a couple of di®erential equations. We also investigate the e®ects of Palatini f(R) terms in the formulation of structure scalars obtained by orthogonal splitting of the Riemann tensor in general relativity. We then identify the parameters which produce energy density irregularities in expansion and expansion-free dissipative as well as non-dissipative matter distributions. It is found that particular combinations of the matter variables lead to irregularities in an initially homogeneous °uid dis- tribution. We conclude that Palatini f(R) extra corrections tend to decrease the inhomogeneity, thereby imparting stability to the self-gravitating system.