  Conformal Carroll symmetry generically arises on null manifolds and is
important for holography of asymptotically flat spacetimes, generic black hole
horizons and tensionless strings. In this paper, we focus on two dimensional
(2d) null manifolds and hence on the 2d Conformal Carroll or equivalently the
3d Bondi-Metzner-Sachs (BMS) algebra. Using Carroll covariance, we write the
most general free massless Carroll scalar field theory and discover three
inequivalent actions. Of these, two viz. the time-like and space-like actions,
have made their appearance in literature before. We uncover a third that we
call the mixed-derivative theory. As expected, all three theories enjoy
off-shell BMS invariance. Interestingly, we find that the on-shell symmetry of
mixed derivative theory is a single Virasoro algebra instead of the full BMS.
We discuss potential applications to tensionless strings and flat holography.
