The modified Camassa-Holm equation (mCH) with a cubic nonlinearity is an integrable and nonlocal mathematical model for the unidirectional propagation of shallow water waves， admitting special explicit solutions such as multi-solitons and breathers. By employing negative ordered conserved quantities, it is demonstrated that multi-solitons and breathers, are orbitally stable under perturbations in high regularity spaces. We also discuss the issue of embedded eigenvalue of the linearized operator at the single soliton, which is related to asymptotic stability.