Can anyone recommend a good overview of singularity theory? In particular, quotient singularities...

7$\begingroup$ Could you provide a little more context? I think your question is too broad to have a useful criterion for an answer. Morse theory can be considered singularity theory, for example, but it's not clear if you want to allow that general a context. $\endgroup$ – Ryan Budney Mar 6 '12 at 2:00

$\begingroup$ Stratified Morse Theory is more useful than Morse Theory for the study of singularities via differential topology. Such applications to algebraic geometry are discussed in the canonical reference, which is a book/monograph called Stratified Morse Theory. (I'm making this a comment because, among other things, I don't know any stratified Morse theory.) $\endgroup$ – Charles Staats Mar 6 '12 at 20:54
A book I found very helpful was the book by János Kollár, named Lectures on Resolution of Singularities. In §2.3, he discusses Quotient Singularities, mostly in the context of resolution for surfaces.
Curves and Singularities by J.W. Bruce and P.J. Giblin gives a highly readable overview of basic singularity theory.
If you are looking for a more topological treatment of the subject, there is a twovolume Singularities of Differentiable Maps by Arnold, Varchenko and GuseinZade.
Perhaps Young person's guide to canonical singularities by Miles Reid. The connection seems to be that at least for surfaces, canonical singularities are exactly quotient singularities by finite subgroups of SL(2,C), (rational double points). In higher dimension the two types no longer coincide. I am a novice, and merely repeating what I have noted from browsing.
Golubitsky and Guillemin Stable Mappings and their singularities is a fairly canonical reference for the ThomMatherLevine thread of ideas, building off of Whitney and Morse's work.
There is the classic Singular Points of Complex Hypersurfaces by J. W. Milnor. Try also Topics in real and complex singularities by A. Dimca.