- Th. Brocker, Differentiable Germs and Catastrophes. CUP. 1975.
- J.W. Bruce & P.J. Giblin, Curves and Singularities. CUP (2nd ed 1992)
- T. Poston & I. Stewart, Catastrophe Theory and Applications. Dover, 1996. (Original edition Prentice Hall, 1978).
More advanced books
- C.G. Gibson, Singular Points of Smooth Mappings. Pitman, 1979.
- J. Martinet, Singularities of smooth functions and maps. London Math. Soc. Lecture Note Series 58, Cambridge, 1982
- V. Arnold, V. Goryunov, O.V. Lyashko & V.A. Vasil'ev, Singularity Theory I. Springer, 1993.
- V. Arnold, S. Gusein-Zade & A. Varchenko, Singularities of Differentiable Maps, Volume 1. Birkauser, 1985.
- M. Golubitsky & D. Schaeffer, Singularities and Groups in Bifurcation Theory: Vol 1, Springer-Verlag, 1985.
- M. Golubitsky & V. Guillemin, Stable Mappings and Their Singularities, Springer-Verlag, 1974.
- Réné Thom, Structural stability and Morphogenesis. W.A. Benjamin Publishers, 1975. (This is the very influential book that started it all off - the original was published in 1972 in French. It has many deep ideas, and I doubt they have all been fully investigated yet.)
- H. Whitney, On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane. Annals of Mathematics, 62 (1955) 374–410. (This is the first research that investigates singularities.)
- V.I. Arnold, Catastrophe Theory. Springer. 1998. A very readable book. (Vladimir Arnold was one of the most influential mathematicians of the second half of the 20th century, and made considerable and fundamental contributions to the understanding and applications of singularity theory.)
- M. Demazure, Bifurcations and Catastrophes: Geometry of Solutions to Nonlinear Problems. Springer (Universitext series), 1999.
- R. Gilmore, Catastrophe Theory for Scientists and Engineers. Dover.
- P.T. Saunders, An Introduction to Catastrophe Theory. CUP, 1980. (This is a nice book which describes Catastrophe Theory and gives many applications, but does not prove any of the theorems; it is written more for engineers or applied scientists than mathematicians.)
For differential equations and flows, I recommend
- V.I. Arnold, Ordinary Differential Equations, MIT Press, 1978.
Material on the web