The polyhedra on this page are not quite regular, but as they are close I present them here as 'near misses'.
These polyhedra are also presented as 'Stress Maps', a bespoke form of VRML model for showing distortion in such models. More information on Stress Maps and distortion statistics is available here.
Johnson Solid Near Misses When Norman Johnson was cataloging the convex polyhedra in the 1960's, he apparently came across "a number of tantalisingly close misses" ... A sequence of the 'nearestmisses' discovered to date is given on this page.
Acrohedral
Near Misses: Attempts
to resolve as yet unsolved acrons.


Enneagonal and Hendecagonal
Polyhedra: Polyhedra containing enneagons or
hendecagons
Dodecagonal Polyhedra: Polyhedra containing dodecagons
Near Misses based on dodecahedra: Some near misses that have a dodecahedral origin.
'Wedges' Near misses formed by attaching an ngon to an (n+1)gon
Rings of antiprisms:
A ring can
be made of thirteen square antiprisms. It
is not quite regular. It's not the closest such near
miss though ...