We consider a singular Sturm—Liouville expression with the indefinite weight sgn x. There is a self-adjoint operator in some Krein space associated naturally with this expression. We characterize the local definitizability of this operator in a neighbourhood of ∞. Moreover, in this situation, the point ∞ is a regular critical point. We construct an operator A = (sgn x)(−d2/dx2 + q) with non-real spectrum accumulating to a real point. The results obtained are applied to several classes of Sturm—Liouville operators.