  We study several classes of isolated singularities of plurisubharmonic
functions that can be approximated by analytic singularities with control over
their residual Monge--Amp\`ere masses. They are characterized in terms of Green
functions for Demailly's approximations, relative types, and valuations.
Furthermore, the classes are shown to appear when studying graded families of
ideals of analytic functions and the corresponding asymptotic multiplier
ideals.
