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.
You could have a look at the nice textbook by S. Ishii Introduction to Singularities. Chapter 7 is devoted to normal two-dimensional 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.