  Recent findings of new Higgs modes in unconventional superconductors require
a classification and characterization of the modes allowed by nontrivial gap
symmetry. Here we develop a theory for a tailored nonequilibrium quantum quench
to excite all possible oscillation symmetries of a superconducting condensate.
We show that both a finite momentum transfer and quench symmetry allow for an
identification of the resulting Higgs oscillations. These serve as a
fingerprint for the ground state gap symmetry. We provide a classification
scheme of these oscillations and the quench symmetry based on group theory for
the underlying lattice point group. For characterization, analytic calculations
as well as full scale numeric simulations of the transient optical response
resulting from an excitation by a realistic laser pulse are performed. Our
classification of Higgs oscillations allows us to distinguish between different
symmetries of the superconducting condensate.
