It is well known that the orbit of a lattice in hyperbolic n-space is uniformly distributed when projected radially onto the unit sphere. I consider the fine-scale statistics of the projected lattice points and express the limit distributions in terms of random hyperbolic lattices. This provides in particular a new perspective on recent results by Boca, Popa, and Zaharescu on 2-point correlations for the modular group, and by Kelmer and Kontorovich for general lattices in dimension n=2. The results are markedly different from the analogues for Euclidean lattices, where fine-scale statistics have been analyzed by Marklof and Strombergsson.  Joint work with Jens Marklof.