Let M be a compact hyperbolic manifold. The entropy bounds of Anantharaman et al. restrict the possible invariant measures on T1M that can be quantum limits of sequences of eigenfunctions. Weaker versions of the entropy bounds also apply to approximate eigenfuctions ("log-scale quasimodes"), so it is interesting to construct such approximate eigenfunctions which converges to singular measures. Generalizing work of Brooks (hyperbolic surfaces) and Eswarathasan--Nonnenmacher (hyperbolic geodesics on Riemannian surfaces) we construct sequences of quasimodes on M converging to totally geodesic submanifolds. A diagonal argument then realizes every invariant measure are a limit of quasimodes of fixed logarithmic width. Joint work with S. Eswarathasan.