  The cutoff phenomenon describes a sharp transition in the convergence of a
Markov chain to equilibrium. In recent work, the authors established cutoff and
its location for the stochastic Ising model on the $d$-dimensional torus
$(Z/nZ)^d$ for any $d\geq 1$. The proof used the symmetric structure of the
torus and monotonicity in an essential way.
  Here we enhance the framework and extend it to general geometries, boundary
conditions and external fields to derive a cutoff criterion that involves the
growth rate of balls and the log-Sobolev constant of the Glauber dynamics. In
particular, we show there is cutoff for stochastic Ising on any sequence of
bounded-degree graphs with sub-exponential growth under arbitrary external
fields provided the inverse log-Sobolev constant is bounded. For lattices with
homogenous boundary, such as all-plus, we identify the cutoff location
explicitly in terms of spectral gaps of infinite-volume dynamics on half-plane
intersections. Analogous results establishing cutoff are obtained for
non-monotone spin-systems at high temperatures, including the gas hard-core
model, the Potts model, the anti-ferromagnetic Potts model and the coloring
model.
