  The notion of singular reduction operators, i.e., of singular operators of
nonclassical (conditional) symmetry, of partial differential equations in two
independent variables is introduced. All possible reductions of these equations
to first-order ODEs are are exhaustively described. As examples, properties of
singular reduction operators of (1+1)-dimensional evolution and wave equations
are studied. It is shown how to favourably enhance the derivation of
nonclassical symmetries for this class by an in-depth prior study of the
corresponding singular vector fields.
