  We present "sheet+release" simulations that reliably follow the evolution of
dark matter structure at and below the dark matter free-streaming scale, where
instabilities in traditional N-body simulations create a large population of
spurious artificial haloes. Our simulations sample a large range of
power-spectrum cutoff functions, parameterized through the half-mode scale
$k_{\rm{hm}}$ and a slope parameter $\beta$. This parameter space can represent
many non-cold dark matter models, including thermal relic warm dark matter,
sterile-neutrinos, fuzzy dark matter, and a significant fraction of ETHOS
models. Combining these simulations with additional N-body simulations, we find
the following results. (1) Even after eliminating spurious haloes, the halo
mass function in the strongly suppressed regime ($n_{\rm{X}}/n_{\rm{CDM}} <
5\%$) remains uncertain because it depends strongly on the definition of a
halo. At these mass scales traditional halo finders primarily identify
overdensities that are unbound, highly elongated, dominated by tidal fields, or
far from virialized. (2) The regime where the suppression is smaller than a
factor of 20 is quite robust to these uncertainties, however, and can be
inferred reliably from suitable N-body simulations. (3) Parameterizing the
suppression in the halo- and subhalo mass functions through the scales where
the suppression reaches $20\%$, $50\%$ and $80\%$, we provide simple formulae
which enable predictions for many non-cold dark matter models. (4) The halo
mass-concentration relations in our sheet+release simulations agree well with
previous results based on N-body simulations. (5) In general, we confirm the
validity of previous N-body studies of warm dark matter models, largely
eliminating concerns about the effects of artificial haloes.
