  In one-dimensional anharmonic lattices, we construct nonlinear standing waves
(SWs) reducing to harmonic SWs at small amplitude. For SWs with spatial
periodicity incommensurate with the lattice period, a transition by breaking of
analyticity versus wave amplitude is observed. As a consequence of the
discreteness, oscillatory linear instabilities, persisting for arbitrarily
small amplitude in infinite lattices, appear for all wave numbers Q not equal
to zero or \pi. Incommensurate analytic SWs with |Q|>\pi/2 may however appear
as 'quasi-stable', as their instability growth rate is of higher order.
