We have investigated the dynamics of a phase-modulated driven gyroscope, including vibrational resonance (VR) phenomena and the stability of the system, both with and without the phase modulation. For appropriate choices of parameters, three distinct equilibrium states can be attained. We use the method of direct separation of the fast and slow motions to show that, in the presence of an additive external driving force, time-periodic phase-modulation promotes resonance suppression. Both giant VR and anti-resonance induction in the quiescent state arise within certain parameter ranges. The implications of the giant and suppressed resonances for gyroscope applications and beyond are discussed.