Properly modulated flat-band lattices have a divergent density of states at the flat-band energy. Quasiperiodic
modulations are known to host a metal-insulator transition already in one space dimension. Their embedding into
flat-band geometries consequently allows for a precise engineering and fine tuning of mobility edges. We obtain
analytic expressions for singular mobility edges for two flat-band lattice examples. In particular, we engineer
cases with arbitrarily small energy separations of mobility edge, zeroes, and divergencies.