We propose lattice soleakons: self-trapped waves that selfconsistently populate slowly-attenuating leaky modes of their self-induced defects in periodic potentials. Two types, discrete and Bragg, lattice soleakons are predicted. Discrete soleakons that are supported by combination of self-focusing and self-defocusing nonlinearities propagate robustly for long propagation distances. They eventually abruptly disintegrate because they emit power to infinity at an increasing pace. In contrast, Bragg soleakons self-trap by only self-focusing nonlinearity. Also, they do not disintegrate because they emit power at a decreasing rate.