  There is a common understanding in quantum optics that nonclassical states of
light are states that do not have a positive semidefinite and sufficiently
regular Glauber-Sudarshan $P$ function. Almost all known nonclassical states
have $P$ functions that are highly irregular which makes working with them
difficult and direct experimental reconstruction impossible. Here we introduce
classes of nonclassical states with regular, non-positive-definite $P$
functions. They are constructed by "puncturing" regular smooth positive $P$
functions with negative Dirac-delta peaks, or other sufficiently narrow smooth
negative functions. We determine the parameter ranges for which such punctures
are possible without losing the positivity of the state, as well as the regimes
yielding antibunching of light. Finally, we propose some possible experimental
realizations of such states.
