A theoretical study on the nonlinear propagation of nonplanar(cylindrical and spherical) electrostatic modified ion-acoustic(mIA) shock structures has been carried out in an unmagnetized,collisionless four component degenerate plasma system(containing degenerate electron fluids,inertial positively as well as negatively charged light ions,and positively charged static heavy ions).This investigation is valid for both non-relativistic and ultra-relativistic limits.The modified Burgers(mB) equation has been derived by employing the reductive perturbation method,and used to numerically analyze the basic features of shock structures.It has been found that the effects of degenerate pressure and number density of electron and inertial positively as well as negatively charged light ion fluids,and various charging state of positively charged static heavy ions significantly modify the basic features of mIA shock structures.The implications of our results to dense plasmas in astrophysical compact objects(e.g.,non-rotating white dwarfs,neutron stars,etc.) are briefly discussed. A theoretical study on the nonlinear propagation of nonplanar (cylindrical and spherical) electrostatic modified ion-acoustic (mIA) shock structures has been carried out in an unmagnetized, collisionless four component degenerate plasma system (containing degenerate electron fluids, inertial positively as well as negatively charged light ions, and positively charged static heavy ions). This investigation is valid for both non-relativistic and ultra-relativistic limits. The modified Burgers (mB) equation has been derived by employing the reductive perturbation method, and used to numerically analyze the basic features of shock structures. It has been found that the effects of degenerate pressure and number density of electron and inertial positively as well as negatively charged light ion fluids, and various charging state of positively charged static heavy ions. shock structures. These implications of our results to dense plasmas in astrophysical compact objects (e.g., non-rotating white dwarfs, neutron stars, etc.) are briefly discussed.