  We start the investigation of free boundary variational models featuring
oscillatory singularities. The theory varies widely depending upon the nature
of the singular power $\gamma(x)$ and how it oscillates. Under a mild
continuity assumption on $\gamma(x)$, we prove the optimal regularity of
minimizers. Such estimates vary point-by-point, leading to a continuum of free
boundary geometries. We also conduct an extensive analysis of the free boundary
shaped by the singularities. Utilizing a new monotonicity formula, we show that
if the singular power $\gamma(x)$ varies in a $W^{1,n^{+}}$ fashion, then the
free boundary is locally a $C^{1,\delta}$ surface, up to a negligible singular
set of Hausdorff co-dimension at least $2$.
