  We study the statistical properties of the sum $S_t=\int_{0}^{t}dt'
\sigma_{t'}$, that is the difference of time spent positive or negative by the
spin $\sigma_{t}$, located at a given site of a $D$-dimensional Ising model
evolving under Glauber dynamics from a random initial configuration. We
investigate the distribution of $S_{t}$ and the first-passage statistics
(persistence) of this quantity. We discuss successively the three regimes of
high temperature ($T>T_{c}$), criticality ($T=T_c$), and low temperature
($T<T_{c}$). We discuss in particular the question of the temperature
dependence of the persistence exponent $\theta$, as well as that of the
spectrum of exponents $\theta(x)$, in the low temperature phase. The
probability that the temporal mean $S_t/t$ was always larger than the
equilibrium magnetization is found to decay as $t^{-\theta-\frac12}$. This
yields a numerical determination of the persistence exponent $\theta$ in the
whole low temperature phase, in two dimensions, and above the roughening
transition, in the low-temperature phase of the three-dimensional Ising model.
