  We show that any simply connected topological closed $4$-manifold punctured
along any compact, totally disconnected tame subset $\Lambda$ admits a
continuum of smoothings which are not diffeomorphic to any leaf of a $C^{1,0}$
codimension one foliation on a compact manifold. This includes the remarkable
case of $S^4$ punctured along a tame Cantor set. This is the lowest reasonable
regularity for this realization problem. These results come from a new
criterion for nonleaves in $C^{1,0}$ regularity. We also include a new
criterion for nonleaves in the $C^2$-category.
  Some of our smooth nonleaves are "exotic", i.e., homeomorphic but not
diffeomorphic to leaves of codimension one foliations on a compact manifold.
