  We perform a detailed and illustrative study of the production of keV sterile
neutrino Dark Matter (DM) by decays of singlet scalars in the early Universe.
In the current study we focus on providing a clear and general overview of this
production mechanism. For the first time we study all regimes possible on the
level of momentum distribution functions, which we obtain by solving a system
of Boltzmann equations. These quantities contain the full information about the
production process, which allows us to not only track the evolution of the DM
generation but to also take into account all bounds related to the spectrum,
such as constraints from structure formation or from avoiding too much dark
radiation. In particular we show that this simple production mechanism can,
depending on the regime, lead to strongly non-thermal DM spectra which may even
feature more than one peak in the momentum distribution. These cases could have
particularly interesting consequences for cosmological structure formation, as
their analysis requires more refined tools than the simplistic estimate using
the free-streaming horizon. Here we present the mechanism including all
concepts and subtleties involved, for now using the assumption that the
effective number of relativistic degrees of freedom is constant during DM
production, which is applicable in a significant fraction of the parameter
space. This allows us to derive analytical results to back up our detailed
numerical computations, thus leading to the most comprehensive picture of keV
sterile neutrino DM production by singlet scalar decays that exists up to now.
