  Numerous non-classicality criteria based on the probabilities of experimental
photocount or theoretical photon-number distributions are derived using several
approaches. Relations among the derived criteria are revealed and the
fundamental criteria are identified. They are grouped into parametric systems
that allow the analysis of the non-classicality from different points of view
('local' non-classicality, pairwise character of photon correlations, etc.).
Considering their structure, the criteria may be divided into groups that
differ in the power to resolve the non-classicality. Quantification of the
non-classicality using the Lee non-classicality depth and the non-classicality
counting parameter is discussed. The used number of field's modes is identified
as an important parameter that may cause unexpected results. An appropriate
linear transformation of a photocount (photon-number) distribution into its
s-ordered form, needed for the determination of the non-classicality depth, is
derived. For comparison, the derived criteria are applied both to an
experimental photocount histogram of a twin beam and the reconstructed
photon-number distributions with different levels of the noise.
