We consider a broad class of nonlinear elliptic equations in a punctured domain and give a complete classification of the behaviour near an isolated singularity for all positive solutions. An important feature of our study lies in the incorporation of inverse square potentials and weighted nonlinearities, whose asymptotic behaviour is modeled by regularly varying functions. In particular, we find sharp conditions such that the singularity is removable for all non-negative solutions, thus resolving an open question of Vazquez and Veron (1985).