Is there any "standard" reference for (rational) singularities on algebraic surfaces? I'm aware of Artin's papers and the one of Brieskorn (Rationale Singularitäten komplexer Flächen), but they seem pretty dense to me. I'm looking for a more extensive treatment using consistent notation, ideally (chapters of) a book.

2$\begingroup$ have you looked at chapter 4, Singularities and Surfaces, in Miles Reid's contribution, "Chapters on Algebraic Siurfaces", to the IAS/Park City mathematics series vol. 3, Complex Algebraic Geometry, edited by Kolla'r? He discusses rational and elliptic Gorenstein singularities, based on the standard papers. $\endgroup$ – roy smith Apr 25 '16 at 17:08

$\begingroup$ @roy smith thanks for the reference, do you know more texts like this one? $\endgroup$ – user269218 Apr 25 '16 at 20:54

1$\begingroup$ There is some good stuff in Chapters 3 and 4 (I think) of Badescu's book. There is also Lipman's classic paper. $\endgroup$ – Hoot Apr 25 '16 at 21:47

$\begingroup$ @Hoot Badescu's detailed proof of the classification of rational double points is nice. Do you know anything similar for rational triple points? Artin doesn't really prove his classification (as far as I can see), he just gives the list. $\endgroup$ – user269218 Apr 26 '16 at 10:19

1$\begingroup$ From a analytic point of view, you may want to check out the book "Normal TwoDimensional Singularities" by Henry B. Laufer. $\endgroup$ – Fei YE Apr 28 '16 at 1:46
You could have a look at the nice textbook by S. Ishii Introduction to Singularities. Chapter 7 is devoted to normal twodimensional singularities, with an extensive treatment of rational ones.
You might be also interested in T. Okuma's book Plurigenera of Surface Singularities. Many topics on rational singularities are discussed in Chapter 2.