  We introduce a setting based on the one-dimensional (1D) nonlinear
Schroedinger equation (NLSE) with the self-focusing (SF) cubic term modulated
by a singular function of the coordinate, |x|^{-a}. It may be additionally
combined with the uniform self-defocusing (SDF) nonlinear background, and with
a similar singular repulsive linear potential. The setting, which can be
implemented in optics and BEC, aims to extend the general analysis of the
existence and stability of solitons in NLSEs. Results for fundamental solitons
are obtained analytically and verified numerically. The solitons feature a
quasi-cuspon shape, with the second derivative diverging at the center, and are
stable in the entire existence range, which is 0 < a < 1. Dipole (odd) solitons
are found too. They are unstable in the infinite domain, but stable in the
semi-infinite one. In the presence of the SDF background, there are two
subfamilies of fundamental solitons, one stable and one unstable, which exist
together above a threshold value of the norm (total power of the soliton). The
system which additionally includes the singular repulsive linear potential
emulates solitons in a uniform space of the fractional dimension, 0 < D < 1. A
two-dimensional extension of the system, based on the quadratic nonlinearity,
is formulated too.
