  We extend basic regularity of the free boundary of the obstacle problem to
some classes of heterogeneous quasilinear elliptic operators with variable
growth that includes, in particular, the $p(x)$-Laplacian. Under the assumption
of Lipschitz continuity of the order of the power growth $p(x)>1$, we use the
growth rate of the solution near the free boundary to obtain its porosity,
which implies that the free boundary is of Lebesgue measure zero for
$p(x)$-Laplacian type heterogeneous obstacle problems.
  Under additional assumptions on the operator heterogeneities and on data we
show, in two different cases, that up to a negligible singular set of null
perimeter the free boundary is the union of at most a countable family of $C^1$
hypersurfaces:
  i) by extending directly the finiteness of the $(n-1)$-dimensional Hausdorff
measure of the free boundary to the case of heterogeneous $p$-Laplacian type
operators with constant $p$; $1<p<\infty$;
  ii) by proving the characteristic function of the coincidence set is of
bounded variation in the case of non degenerate or non singular operators with
variable power growth $p(x)>1$.
