Interlaminar and intralaminar failure modes of composite conoidal shells are investigated in this paper using finite element as the mathematical tool. One of the prerequisites for confidently applying the composite shells in practical civil engineering is to have a comprehensive idea about the shell behavior under different loading conditions. The static bending and free vibration responses of laminated composite conoidal shells are now well known as a good number of technical papers have been published by different researchers. But for these materials which are weak in transverse shear failure may initiate at any inner lamina or interface and may remain undetected and unattended. Such latent damages may progress gradually and lead to a sudden total ply failure. Due to fabrication problems or overloading, delamination defect may occur and remain hidden within the lamina. Hence it is imperative to have knowledge of the first ply failure load and the load at which delamination initiates. To explore this area, in this paper, a simply supported shell is considered loaded by concentrated load. Lamination and the load position are varied to investigate the first ply failure characteristics of the composite shell. Well accepted failure theories like maximum stress, Hoffman, Tsai-Wu and Tsai-Hill are used in the current formulation to obtain failure loads. While obtaining the failure loads, each lamina is considered to be under plane stress condition i.e. no transverse stresses are considered to act on any lamina. Comparison of failure loads obtained from both the failure modes are also presented to get the minimum failure load which will govern the shell design.