  The rare-earth perovskite TbScO$_3$ has been widely used as a substrate for
the growth of epitaxial ferroelectric and multiferroic thin films, while its
detailed low-temperature magnetic properties were rarely reported. In this
paper, we performed detailed magnetization, specific heat and single crystal
neutron scattering measurements, along with the crystalline electric field
calculations to study the low-temperature magnetic properties of TbScO$_3$. All
our results suggest the magnetic Tb$^{3+}$ has an Ising-like pseudo-doublet
ground state at low temperatures. Due to the constrain of local point symmetry,
these Tb$^{3+}$ Ising moments are confined in the $ab$ plane with a tilt angle
of $\varphi = \pm48^{\mathrm{o}}$ to the $a$ axis. In zero field, the system
undergoes an antiferromagnetic phase transition at $T_{\mathrm{N}}=2.53$ K, and
forms a $G_xA_y$ noncollinear magnetic structure below $T_{\mathrm{N}}$. We
find the dipole-dipole interactions play an important role to determine the
magnetic ground state, which are also responsible for the quasi-one-dimensional
magnetism in TbScO$_3$. The significant anisotropic diffuse scatterings further
confirm the quasi-one-dimensional magnetism along the $c$ axis. The magnetic
phase diagram with the field along the easy $b$ axis is well established. In
addition to the $G_xA_y$ antiferromagnetic state, there is an exotic
field-induced phase emerged near the critical field $B_{\mathrm{c}}\simeq0.7$
T, where three-dimensional magnetic order is suppressed but strong
one-dimensional correlations may still exist.
