The stability of the membranous initial stress-strain state of equilibrium of thin elastic shells is studied. Using the two-dimensional theory of the Kirchhof-Love type, the linear problem of equilibrium bifurcation is considered. Buckling forms, localized near some lines or points on the middle surface of shell, called the weakest lines or points are studied. The asymptotic integration methods for equations with a parameter developed by A.L. Goldenveiser and V.P. Maslov are used. Approximate formulas for critical loads buckling forms which decrease with the deviation from the weakest line or point are found.