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The Eddington-inspired-Born-Infeld scenario (EiBI) can prevent the Big Bangsingularity for a matter content whose equation of state is constant andpositive. In a recent paper we showed that, on the contrary, it is impossibleto smooth a big rip in the EiBI setup. In fact the situations are stilldifferent for other singularities. In this paper we show that a big freezesingularity in general relativity (GR) can in some cases be smoothed to asudden or a type IV singularity under the EiBI scenario. Similarly, a sudden ora type IV singularity in GR can be replaced in some regions of the parameterspace by a type IV singularity or a loitering behaviour, respectively, in theEiBI framework. Furthermore, we find that the auxiliary metric related to thephysical connection usually has a smoother behaviour than that based on thephysical metric. In addition, we show that bound structures close to a big ripor a little rip will be destroyed before the advent of the singularity and willremain bound close to a sudden, big freeze or type IV singularity. We thenconstrain the model following a cosmographic approach, which is well-known tobe model-independent, for a given Friedmann-Lema\^itre-Robertson-Walkergeometry. It turns out that among the various past or present singularities,the cosmographic analysis can pick up the physical region that determines theoccurrence of a type IV singularity or a loitering effect in the past.Moreover, to determine which of the future singularities or doomsdays is moreprobable, observational constraints on higher order cosmographic parameters isrequired.

Eddington-Born-Infeld cosmology: a cosmographic approach, a tale of doomsdays and the fate of bound structures