  We study the Plateau problem with a lower dimensional obstacle in
$\mathbb{R}^n$. Intuitively, in $\mathbb{R}^3$ this corresponds to a soap film
(spanning a given contour) that is pushed from below by a "vertical" 2D
half-space (or some smooth deformation of it). We establish almost optimal
$C^{1,1/2-}$ estimates for the solutions near points on the free boundary of
the contact set, in any dimension $n\ge 2$.
  The $C^{1,1/2-}$ estimates follow from an $\varepsilon$-regularity result for
minimal surfaces with thin obstacles in the spirit of the De Giorgi's
improvement of flatness. To prove it, we follow Savin's small perturbations
method. A nontrivial difficulty in using Savin's approach for minimal surfaces
with thin obstacles is that near a typical contact point the solution consists
of two smooth surfaces that intersect transversally, and hence it is not very
flat at small scales. Via a new "dichotomy approach" based on barrier arguments
we are able to overcome this difficulty and prove the desired result.
