  The cosmological principle asserts that the Universe looks spatially
homogeneous and isotropic on sufficiently large scales. Given the fundamental
implications of the cosmological principle, it is important to empirically test
its validity on various scales. In this paper, we use the Type Ia supernova
(SN~Ia) magnitude-redshift relation, from both the Pantheon and JLA
compilations, to constrain theoretically motivated anisotropies in the Hubble
flow. In particular, we constrain the quadrupole moment in the effective Hubble
parameter and the dipole moment in the effective deceleration parameter. We
find no significant quadrupole term regardless of the redshift frame we use.
Our results are consistent with the theoretical expectation of a quadrupole
moment of a few percent at scales of $\sim 100 h^{-1}$ Mpc. We place an upper
limit of a $\sim 10\%$ quadrupole amplitude relative to the monopole, $H_0$, at
these scales. We find that we can detect a $\sim 7\%$ quadrupole moment at the
5$\sigma$ level, for a forecast low-$z$ sample of 1055 SNe~Ia. We find an
exponentially decaying dipole moment of the deceleration parameter varies in
significance depending on the redshift frame we use. In the heliocentric frame,
as expected, it is detected at $\sim 3 \sigma$ significance. In the rest-frame
of the cosmic microwave background (CMB), we find a marginal $\sim 2 \sigma$
dipole, however, after applying peculiar velocity corrections, the dipole is
insignificant. Finally, we find the best-fit frame of rest relative to the
supernovae to differ from that of the CMB.
