We study linear and nonlinear dynamics of strained graphene lattices. We find the region of structural stability of flat graphene sheet in the space of two-component strain (ε_xx, ε_yy) with x and y axes oriented along the zigzag and armchair directions, respectively. We demonstrate that a gap in the phonon spectrum appears when graphene is strained by uniaxial load either along zigzag or along armchair direction, while no gap appears for hydrostatic loading. We find that discrete breathers (DBs) can be generated when graphene is uniaxially loaded in zigzag direction, and the DB frequency decreases with the amplitude revealing soft-type anharmonicity. An unusual feature of the DBs observed in this system is that they can have the frequencies within the phonon spectrum. This can be explained by the fact that DBs are polarized along the armchair direction, while the band of phonon spectrum containing DB frequency is occupied by the phonons having only out-of-plane atomic displacements. DB does not radiate energy because even large in-plane displacements of atoms in flat graphene are only weakly coupled with out-of-plane displacements.