The formation and propagation of nonlinear ion-acoustic (IA) waves are studied in an unmagnetized cold viscous plasma, comprising the inertial ions and superthermal trapped electrons in cylindrical geometry with transverse perturbations. The well-known reductive perturbation method is employed to derive cylindrical trapped Kadomtsev Petviashvili Burgers (CTKPB) and cylindrical trapped Kadomtsev Petviashvili (CTKP) equations with and without dissipation effects, respectively. The exact solutions of CTKPB and CTKP equations are obtained for the first time by utilizing the dependent variable transformation. The soliton and shock structures are found to be significantly affected by the plasma parameters including the trapping efficiency parameter β, the superthermality parameter κ, viscosity µ0 and the geometrical effects. Importantly, the geometrical effects and transverse perturbations alter the shape of solitons and shocks, resulting into the parabolic nonlinear structures. These structures become more parabolic at later times due to an interplay of transverse and time coordinates (η, τ). The results of the present study might be helpful to understand the characteristics of IA structures in space plasmas, such as auroral regions, where energetic trapped electrons have been observed.