So a friend is doing his PhD in maths (algebraic topology) and his advisor wants him to publish something on singularities (of which, as fas as I understand, he knows next to nothing). I want to give him for his birthday a book on the topic, something general but advanced (the basics he can get from a hundred places), perhaps research oriented. I don't know, a sort of reference book, or whatever you consider to be THE book on singularities.
Thanks
There is always the two volume classic by Arnold, Guisein-Zade, and Varchenko. Here are the Springer pages for both: Volume 1 and Volume 2. I've only looked at volume 1 but it is very good. As far as I know, these books are "THE books" on the subject though I'm not an expert.
There is also this very nice book by Greuel, Lossen, and Shustin that takes a more complex algebraic approach and has a focus on deformations of singularities. It is very example based which is great for a lot of readers. A google search reveals the full book linked here.
Finally, I would recommend Dimca's book Sheaves in Topology. The focus isn't exactly on singularities but instead it is focused on giving a relatively quick introduction to the main tools used in the modern study of the topology of singularities, mainly without proofs, and then has the last chapter devoted to showing how one applies these tools to specific problems about singularities. As I said, this book doesn't have proofs of the main theorems because it is meant to give all the tools needed as fast as possible so maybe best fits your description of a reference book for the working researcher.