  In this work we extend the $L^1$-Bj\"ork-Sj\"olin theory of strongly singular
convolution operators to arbitrary graded Lie groups. Our criteria are
presented in terms of the oscillating H\"ormander condition due to Bj\"ork and
Sj\"olin of the kernel of the operator, and the decay of its group Fourier
transform is measured in terms of the infinitesimal representation of an
arbitrary Rockland operator. The historical result by Bj\"ork and Sj\"olin is
re-obtained in the case of the Euclidean space.
