  Variational inequalities with thin obstacles and Signorini-type boundary
conditions are classical problems in the calculus of variations, arising in
numerous applications. In the linear case many refined results are known, while
in the nonlinear setting our understanding is still at a preliminary stage.
  In this paper we prove $C^1$ regularity for the solutions to a general class
of quasi-linear variational inequalities with thin obstacles and $C^{1,
\alpha}$ regularity for variational inequalities under Signorini-type
conditions on the boundary of a domain.
