  Existence of a spectral singularity (SS) in the spectrum of {a
Schr\"{o}dinger operator with} a non-Hermitian potential requires exact
matching of parameters of the potential. We provide a necessary and sufficient
condition for a potential to have a SS at a given wavelength. It is shown that
potentials with SSs at prescribed wavelengths can be obtained by a simple and
effective procedure. In particular, the developed approach allows one to obtain
potentials with several SSs and with SSs of the second order, as well as
potentials obeying a given symmetry, say, $\PT-$symmetric potentials. Also, the
problem can be solved when it is required to obtain a potential obeying a given
symmetry, say, $\PT-$symmetric potential. We illustrate all these opportunities
with examples. We also describe splitting of a second-order SSs under change of
the potential parameters, and discuss possibilities of experimental observation
of SSs of different orders.
