The destabilization of fast magnetoacoustic waves (FMWs) and Alfvén waves (AWs) in plasmas with energetic ions having strongly anisotropic velocity distribution is considered. It is shown that when the ratio of the energetic ion velocity to the Alfvén velocity is large enough, the growth rate of the low-frequency FMW instability has sharp maxima at certain angles of the wave propagation. On the other hand, low-frequency FMWs are always stable, whereas the AWs and high-frequency FMWs can be unstable when the waves propagate along the magnetic field. The developed theory is applied to a plasma of the flashing loops of the Solar Corona. It is found that the presence of a very small amount of the energetic ions can lead to instabilities with the growth rate, which considerably exceeds the inverse characteristic collisional times. It is concluded that the frequency spectrum of the destabilized waves can be rather wide, which may lead to the heating of the bulk plasma ions on the sub-harmonics of the ion gyrofrequency due to the mechanism suggested by Chen et al.