In this article the plain elasticity problem for a semi-strip with a transverse crack is investigated in the different cases of the boundary conditions at the semi-strip’s end. Unlike many works dedicated to this subject, the fixed singularities in the singular integral equation’s kernel are considered. The integral transformations’ method is applied by the generalized scheme to reduce the initial problem to a one-dimensional problem. The one-dimensional problem is formulated as the vector boundary value problem which is solved with the help of matrix differential calculations and Green’s matrix apparatus. The solution of the problem is reduced to the solving of the system of three singular integral equations. Depending on the conditions given on the short edge of the semi-strip, the constructed singular integral equation can have one, or two fixed singularities. A special method is applied to solve this equation in regard of the singularities existence. Hence the system of the singular integral equations (SSIE) is solved with the help of the generalized method. The stress intensity factors (SIF) are investigated for different lengths of crack. The novelty of this work is in the application of new approach allowing the consideration of the fixed singularities in the problem about a transverse crack in the elastic semi-strip. The comparison of the numerical results’ accuracy during the usage of the different approaches to the solving of SSIE is worked out.