  An absolute continuity approach to quasinormality which relates the operator
in question to the spectral measure of its modulus is developed. Algebraic
characterizations of some classes of operators that emerged in this context are
invented. Various examples and counterexamples illustrating the concepts of the
paper are constructed by means of weighted shifts on directed trees.
Generalizations of these results that cover the case of q-quasinormal operators
are established.
