  This paper aims at providing rigorous numerical computation procedure for
finite-time singularities in dynamical systems. Combination of time-scale
desingularization as well as Lyapunov functions validation on stable manifolds
of invariant sets for desingularized vector fields with standard integration
procedure for ordinary differential equations give us validated trajectories of
dynamical systems involving finite-time singularities. Our focus includes
finite-time extinction, traveling wave solutions with half-line or compact
support, and singular canards in fast-slow systems, including rigorous
validations of enclosures of extinction, finite-passage times or size of
supports for compactons. Such validated solutions lead to a plenty of composite
wave solutions for degenerate parabolic equations, for example, with concrete
information of profiles and evolutions. The present procedure also provides a
universal aspect of finite-time singularities with rigorous numerics, combining
with rigorous numerics of blow-up solutions in preceding works.
