We investigate two instabilities of spiral waves in oscillatory media subject to different types of forcing using the complex Ginzburg-Landau equation. First, the transition of spiral waves via so-called superspirals to spatio-temporal chaos is related to a coexistence of the Eckhaus instability of the wave field and the intrinsic oscillatory meandering instability of the spiral core. Second, resonantly forced oscillatory media axe shown to possess a novel scenario of spiral breakup. Bifurcation analysis and linear stability analysis yield explanations for the phenomenology observed by direct simulations.