  We present a numerical study of the dynamics of the one-dimensional Ising
model by applying the large-deviation method to describe ensembles of dynamical
trajectories. In this approach trajectories are classified according to a
dynamical order parameter and the structure of ensembles of trajectories can be
understood from the properties of large-deviation functions, which play the
role of dynamical free-energies. We consider both Glauber and Kawasaki
dynamics, and also the presence of a magnetic field. For Glauber dynamics in
the absence of a field we confirm the analytic predictions of Jack and Sollich
about the existence of critical dynamical, or space-time, phase transitions at
critical values of the "counting" field $s$. In the presence of a magnetic
field the dynamical phase diagram also displays first order transition
surfaces. We discuss how these non-equilibrium transitions in the 1$d$ Ising
model relate to the equilibrium ones of the 2$d$ Ising model. For Kawasaki
dynamics we find a much simple dynamical phase structure, with transitions
reminiscent of those seen in kinetically constrained models.
