  Interest in cosmological singularities has remarkably grown in recent times,
particularly on future singularities with the discovery of late-time
acceleration of the universe and dark energy. While such explorations have
previously been done in various modified gravity and quantum gravitationally
inspired cosmologies (besides standard general relativistic cosmology), no such
an endeavour has been taken up till now in the realms of renormalization group
approaches to cosmology and we have hence took up on this journey. In this
work, we consider the formation of cosmological singularities in an
asymptotically safe cosmology where the cut off scale is proportional to the
Hubble parameter. We consider a well motivated inhomogeneous form of the
equation of state(EOS) as well. We firstly delve into some basics of this
cosmology and show that such a scenario permits a transition between phantom
and quintessence forms of universal evolution. We then show that one can have
Type I - Type IV singularities in such a cosmology for various version of the
EOS and that the singularities can occur both in finite and infinite time. The
conditions in which these singularities occur is significantly different than
how they occur in the standard cosmology, with the formulations being even more
involved. Interestingly this difference comes about without the need of any
free parameters in the cosmological models, which is usually what one observes
when one goes for the same pursuit via various modified gravity/ QG inspired
cosmological approaches. Furthermore, we showed that usual singularity removal
methods like conformal anomaly effects and f(R) gravity effects do not offer
much hope for singularity removal in this cosmology.
