Definition singular manifold I'm looking for the definition of a singular manifold. I haven't found it yet. For instance, in $\mathbb{R}^4$, with $f(x,y,z,t)=xy-zt$, $f^{-1}(0)$ is a singular submanifold. I only found a few examples but not a proper definition.
Does anyone know a definition or a book where I can find one?
 A: The most general theory of singular manifolds that I have seen (staying, however, in the framework where one can still talk about differentiable functions, etc) is Matthias Kreck's "stratifolds". See here. Stratifolds are more general than analytical varieties (mentioned by Jack Lee) but less general then, say, simplicial complexes. (Even in the context of similicial complexes, one can still talk about differential forms and de Rham complex to some extent. This theory was developed by Dennis Sullivan in the early 1970s, with motivation coming from the rational homotopy theory.)  
You also may want to take a look in the book "Stratified Morse theory" by Goresky and McPherson. 
However, you should also ask yourself, what is it that you are actually interested in, since the theory of singular manifolds is usually aimed at something rather specific. For instance,  the book by Goresky and McPherson is aimed at algebro-geometric applications, while the one by Kreck is aimed at developing algebraic topology using differential tools. 
