  We consider optimal stopping problems with finite-time horizon and
state-dependent discounting. The underlying process is a one-dimensional linear
diffusion and the gain function is time-homogeneous and difference of two
convex functions. Under mild technical assumptions with local nature we prove
fine regularity properties of the optimal stopping boundary including its
continuity and strict monotonicity. The latter was never proven with
probabilistic arguments. We also show that atoms in the signed measure
associated with the second order spatial derivative of the gain function induce
geometric properties of the continuation/stopping set that cannot be observed
with smoother gain functions (we call them \emph{continuation bays} and
\emph{stopping spikes}). The value function is continuously differentiable in
time without any requirement on the smoothness of the gain function.
