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We present some examples of detailed analysis of intrinsic localized modes in lattices, using the accurate numerical methods derived from the proof of existence of MacKay-Aubry. We report on some improvements on the methods, which are then used to the fullest to obtain the Floquet analysis of the breather solutions. Such calculations are possible taking into account the whole lattice, without any approximations. This yields an unprecedented detail of the mechanisms that govern instabilities in discrete breathers, giving a complete picture of the interplay between localized (“internal”) and extended (“phonon-like”) instabilities.