  Motivated by the new theoretical paradigm that views spacetime geometry as
emerging from the entanglement of a pre-geometric theory, we investigate the
issue of the signature of the presence of horizons and localized matter on the
entanglement entropy (EE) S_E for the case of three-dimensional AdS (AdS_3)
gravity. We use the holographically dual two-dimensional CFT on the torus and
the related modular symmetry in order to treat bulk black holes and conical
singularities (sourced by pointlike masses not shielded by horizons) on the
same footing. In the regime where boundary tori can be approximated by
cylinders we are able to give universal expressions for the EE of black holes
and conical singularities. We argue that the presence of horizons/localized
matter in the bulk is encoded in the EE in terms of (i) enhancement/reduction
of the entanglement of the AdS_3 vacuum, (ii) scaling as area/volume of the
leading term of the perturbative expansion of S_E, (iii) exponential/periodic
behaviour of S_E, (iv) presence of unaccessible regions in the
noncompact/compact dimension of the boundary cylinder. In particular, we show
that the reduction effect of matter on the entanglement of the vacuum found by
Verlinde for the de Sitter vacuum extends to the AdS_3 vacuum
