We consider generalized spectra in $C^*$-algebras of operators on Hilbert $C^*$-modules and give a description of such spectra of shift operators, unitary, self-adjoint and normal operators on the standard Hilbert $C^*$-module over a unital $C^*$-algebra. Moreover, we proceed further by considering the generalized spectra induced by various subclasses of semi-$C^*$-Fredholm operators and study the relationship between these spectra.