By utilizing the general solutions derived for the plies with arbitrary fiber orientations under uniform axial strain (Huang and Chen, 1994), the explicit solutions of the edge-delamination stress singularities for the angle-ply and cross-ply laminates are obtained. The dominant edge-delamination stress singularities for the angle-ply laminates are found to be a real constant, −1/2, and a pair of complex conjugates, −1/2 ± i/2π ln {(b + b^{2} − a^{2})/a}. For the cross-ply laminates, the significant effect of transverse shear stresses of the laminate is considered and the dominant edge-delamination stress singularities are shown as −1/2 and −1/2 ± i/2π ln {(c^{2} + c^{2}_{2} − 4c_{1}c_{3})/2c_{1}}. a, b, cl , c2 , and c3 are the corresponding combined complex constants. In addition, two elementary forms of edge-delamination stress singularity, say, r−1/2 and r^{δr} (ln r)^{n} (δ_{r} = n − 3/2, n = 1, 2 . . .) exist for both the angle-ply and cross-ply laminates. Excellent correlations between the present results and available solutions show the validity of the approach. The deficiencies of the solutions available in the literature are compensated. New results for other angle-ply and cross-ply laminates are also provided.