  The Eddington-inspired-Born-Infeld scenario (EiBI) can prevent the Big Bang
singularity for a matter content whose equation of state is constant and
positive. In a recent paper we showed that, on the contrary, it is impossible
to smooth a big rip in the EiBI setup. In fact the situations are still
different for other singularities. In this paper we show that a big freeze
singularity in general relativity (GR) can in some cases be smoothed to a
sudden or a type IV singularity under the EiBI scenario. Similarly, a sudden or
a type IV singularity in GR can be replaced in some regions of the parameter
space by a type IV singularity or a loitering behaviour, respectively, in the
EiBI framework. Furthermore, we find that the auxiliary metric related to the
physical connection usually has a smoother behaviour than that based on the
physical metric. In addition, we show that bound structures close to a big rip
or a little rip will be destroyed before the advent of the singularity and will
remain bound close to a sudden, big freeze or type IV singularity. We then
constrain the model following a cosmographic approach, which is well-known to
be model-independent, for a given Friedmann-Lema\^itre-Robertson-Walker
geometry. It turns out that among the various past or present singularities,
the cosmographic analysis can pick up the physical region that determines the
occurrence of a type IV singularity or a loitering effect in the past.
Moreover, to determine which of the future singularities or doomsdays is more
probable, observational constraints on higher order cosmographic parameters is
required.
