  While negativity in phase space is a well-known signature of nonclassicality,
a wide variety of nonclassical states require their characterization beyond
negativity. We establish a framework of nonclassicality in phase space that
addresses nonclassical states comprehensively with a direct experimental
evidence. This includes the negativity of phase-space distribution as a special
case and further analyzes quantum states with positive distributions
effectively. We prove that it detects all nonclassical Gaussian states and all
non-Gaussian states of arbitrary dimension remarkably by examining three
phase-space points only. Our formalism also provides an experimentally
accessible lower bound for a nonclassicality measure based on trace distance.
Importantly, this foundational approach can be further adapted to constitute
practical tests in two directions looking into particle and wave nature of
bosonic systems, via an array of nonideal on-off detectors and coarse-grained
homodyne measurement, respectively. All these tests are practically powerful in
characterizing nonclasssical states reliably against noise, making a versatile
tool for a broad range of quantum systems in quantum technologies.
