  This article provides a focused review of recent findings which demonstrate,
in some cases quite counter-intuitively, the existence of bound states with a
singularity of the density pattern at the center, while the states are
physically meaningful because their total norm converges. One model of this
type is based on the 2D Gross-Pitaevskii equation (GPE) which combines the
attractive potential ~ 1/r^2 and the quartic self-repulsive nonlinearity,
induced by the Lee-Huang-Yang effect (quantum fluctuations around the
mean-field state). The GPE demonstrates suppression of the 2D quantum collapse,
driven by the attractive potential, and emergence of a stable ground state
(GS), whose density features an integrable singularity ~1/r^{4/3} at r --> 0.
Modes with embedded angular momentum exist too, and they have their stability
regions. A counter-intuitive peculiarity of the model is that the GS exists
even if the sign of the potential is reversed from attraction to repulsion,
provided that its strength is small enough. This peculiarity finds a relevant
explanation. The other model outlined in the review includes 1D, 2D, and 3D
GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive
terms, respectively. These equations give rise to stable singular solitons,
which represent the GS for each dimension D, with the density singularity
~1/r^{2/(4-D). Such states may be considered as a result of screening of a
"bare" delta-functional attractive potential by the respective nonlinearity.
