  Oscillating scalar fields are useful to model a variety of matter components
in the universe. One or more scalar fields participate in the reheating process
after inflation, while at much lower energies scalar fields are robust dark
matter candidates. Pertaining structure formation in these models, it is well
known that inhomogeneities of the Klein-Gordon field are unstable above the
characteristic De Broglie wavelength. In this paper we show that such
instability implies the existence of a threshold amplitude for the collapse of
primordial fluctuations. We use this threshold to correctly predict the
cut--off scale of the matter power spectrum in the scalar field dark matter
model. Furthermore, for a Klein-Gordon field during reheating we show that this
same threshold allows for abundant production of structure (oscillons but not
necessarily black holes). Looking at the production of Primordial Black Holes
(PBHs) in this scenario we note that the sphericity condition yields a much
lower probability of PBH formation at the end of inflation. Remarkably, even
after meeting such stringent condition, we find that PBHs may be overproduced
during reheating. We finally constrain the epochs at which an oscillating
Klein-Gordon field could dominate the early universe.
