In the beginning there was Thom. So write Poston and Stewart in their book, referring to René Thom's influential book Structural Stability and Morphogenesis. This was really the beginning of the many applications and theoretical developments of the subject.
In fact the earliest mathematical results describing singularities were due to Hassler Whitney in the 1950s, when he investigated what singular points of maps of the plane look like. (He proved they look like folds and cusps.)
Other key players in the development of the subject:
John Mather laid the mathematical foundations of the subject in a series of 6 very technical research papers published between 1968 and 1971.
Vladimir Arnold gave a long classification of function-germs (up to codimension 10), and showed how the classification is related to the geometry and classification of Lie algebras. This relationship is what gives the singularities their names (such as Ak and Dk).
Christopher Zeeman popularized Catastrophe Theory in the 1970s following Thom, suggesting many possible applications of the subject. The picture on the home page is his.
Martin Golubitsky showed how to use singularity theory (especially contact equivalence) in studying bifurcation problems.