  We prove that Penrose limits of metrics with arbitrary singularities of
power-law type show a universal leading u^{-2}-behaviour near the singularity
provided that the dominant energy condition is satisfied and not saturated. For
generic power-law singularities of this type the oscillator frequencies of the
resulting homogeneous singular plane wave turn out to lie in a range which is
known to allow for an analytic extension of string modes through the
singularity. The discussion is phrased in terms of the recently obtained
covariant characterisation of the Penrose limit; the relation with null
geodesic deviation is explained in detail.
