  In this article we study the behavior of strongly singular integrals
associated to three different, albeit equivalent, quasi-norms on Heisenberg
groups; these quasi-norms give rise to phase functions whose mixed Hessians may
or may not drop rank along suitable varieties. In the particular case of the
Koranyi norm we improve on previous arguments of the second author and obtain
sharp $L^2$ estimates for the associated operators.
