  We present Bell-type tests of nonclassicality and non-Gaussianity for
single-mode fields employing a generalized quasiprobability function. Our
nonclassicality tests are based on the observation that two orthogonal
quadratures in phase space (position and momentum) behave as independent
realistic variables for a coherent state. Taking four (three) points at the
vertices of a rectangle (right triangle) in phase space, our tests detect every
pure nonclassical Gaussian state and a range of mixed Gaussian states. These
tests also set an upper bound for all Gaussian states and their mixtures, which
thereby provide criteria for genuine quantum non-Gaussianity. We optimize the
non-Gaussianity tests by employing a squeezing transformation in phase space
that converts a rectangle (right triangle) to a parallelogram (triangle), which
enlarges the set of non-Gaussian states detectable in our formulation. We
address fundamental and practical limits of our generalized phase-space tests
by looking into their relation with decoherence under a lossy Gaussian channel
and their robustness against finite data and nonoptimal choice of phase-space
points. Furthermore, we demonstrate that our parallelogram test can identify
useful resources for nonlocality testing in phase space.
