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Abstract We consider the problem of extracting a few desired eigenpairs of the buckling eigenvalue problem , whereKis symmetric positive semi‐definite,K_Gis symmetric indefinite, and the pencil is singular, namely,KandK_Gshare a nontrivial common nullspace. Moreover, in practical buckling analysis of structures, bases for the nullspace ofKand the common nullspace ofKandK_Gare available. There are two open issues for developing an industrial strength shift‐invert Lanczos method: (1) the shift‐invert operator does not exist or is extremely ill‐conditioned, and (2) the use of the semi‐inner product induced byKdrives the Lanczos vectors rapidly toward the nullspace ofK, which leads to a rapid growth of the Lanczos vectors in norms and causes permanent loss of information and the failure of the method. In this paper, we address these two issues by proposing a generalized buckling spectral transformation of the singular pencil and a regularization of the inner product via a low‐rank updating of the semi‐positive definiteness ofK. The efficacy of our approach is demonstrated by numerical examples, including one from industrial buckling analysis.