  We define twelve variants of a Reifenberg's affine approximation property,
which are known to be connected with the singular sets of minimal surfaces.
With this motivation we investigate the regularity of the sets possessing
these. We classify the properties with respect to whether $j$-dimensional
Hausdorff dimension, locally finite $j$-dimensional Hausdorff measure or
countable $j$-rectifiability hold. In showing that varying levels of regularity
hold for the differing properties, quasi-self-similar sets, interesting in
their own right, are constructed as counter examples. These counter examples
also admit a connection to number theory via the use of the normal number
theorem. Additionally, the intriguing result that such complexity in the
counter examples is actually a necessity is shown.
