  In the present paper, we study some generalized Monge--Amp\`ere equations in
terms of special exterior differential systems on a jet space. Moreover, we
construct geometric singular solutions of the generalized Monge--Amp\`ere
equations by using the method of Cauchy characteristics. Furthermore, we give
criteria for geometric singular solutions to be right-left equivalent to the
cuspidal edge, swallowtail and butterfly.
