  We investigate the approximate j-dimensionality of the singularity sets of
minimal surfaces prescribed by Simon. This leads to the clasification of 8
variations of approximately j-dimensional surfacs in terms of dimension and
locally finite Hausdorff measure. We show that the singularity sets must either
be well behaved or essentially purely unrectifiable.
  Examples of the unrectifiable sets that could occur are constructed and
generalised. They are shown to be pseudo-fractal and related to the well known
Koch Sets. Various representations of the measure and dimension of the sets are
shown, as well as a fine balance in the spiralling nature of the sets.
