  The group of singular elements was first introduced by Helmut Hasse and later
it has been studied by numerous authors including such well known
mathematicians as: Cassels, Furtw\"{a}ngler, Hecke, Knebusch, Takagi and of
course Hasse himself; to name just a few. The aim of the present paper is to
present algorithms that explicitly construct groups of singular and
$S$-singular elements (modulo squares) in a global function field.
