  We investigate the boundedness of oscillating singular integrals on Lie
groups of polynomial growth in order to extend the classical oscillating
conditions due to Fefferman and Stein for the boundedness of oscillating
convolution operators. Kernel criteria are presented in terms of a fixed
sub-Riemannian structure on the group induced by a sub-Laplacian associated to
a H\"ormander system of vector fields. In the case where the group is graded,
kernel criteria are presented in terms of the Fourier analysis associated to an
arbitrary Rockland operator.
