  The classical pitchfork of singularity theory is a twice-degenerate
bifurcation that typically occurs in dynamical system models exhibiting Z_2
symmetry. Non-classical pitchfork singularities also occur in many
non-symmetric systems, where the total bifurcation environment is usually more
complex. In this paper three-dimensional manifolds of critical points, or
limit-point shells, are introduced by examining several bifurcation problems
that contain a pitchfork as an organizing centre. Comparison of these surfaces
shows that notionally equivalent problems can have significant positional
differences in their bifurcation behaviour. As a consequence, the parameter
range of jump, hysteresis, or phase transition phenomena in dynamical models
(and the physical systems they purport to represent) is determined by other
singularities that shape the limit-point shell.
