  We address detection of quantum non-Gaussian states, i.e. nonclassical states
that cannot be expressed as a convex mixture of Gaussian states, and present a
method to derive a new family of criteria based on generic linear functionals.
We then specialise this method to derive witnesses based on $s$-parametrized
quasiprobability functions, generalising previous criteria based on the Wigner
function. In particular we discuss in detail and analyse the properties of
Husimi Q-function based witnesses and prove that they are often more effective
than previous criteria in detecting quantum non-Gaussianity of various kinds of
non-Gaussian states evolving in a lossy channel.
