The 1+d dimensional focussing cubic Schroedinger equation can develop
singular solutions for d>=2. A nonlinear power law dissipative term with
small strength parameter B and power q, modelling multi-photon
absorption, will regularize this singularity at least when this power is
greater than cubic (q>3).

As B approaches zero two possible phenomena are convergence of the
regularised solutions to a singular weak solution with persistent point
singularity and, for q<=3, a threshold below which the regularisation
fails. Numerical simulation can be enlightening but requires special
care due to the extremely small scales that arise in these near-singular
solutions.

Numerical methods and results will be presented which confirm some
conjectures and suggest revisions of others, along with some possible
ingredients for the explanation of why the newly observed behaviour.