  We establish the existence, uniqueness, and $W^{1,2,p}$-regularity of
solutions to fully nonlinear parabolic obstacle problems when the obstacle is
the pointwise supremum of arbitrary functions in $W^{1,2,p}$ and the operator
is only assumed to be measurable in the state and time variables. The results
hold for a large class of non-smooth obstacles, including all convex obstacles.
Applied to stopping problems, they imply that the decision maker never stops at
a convex kink of the stopping payoff. The proof relies on new
$W^{1,2,p}$-estimates for obstacle problems where the obstacle is the maximum
of finitely many functions in $W^{1,2,p}$.
