Most of the responses are pushing you toward the proper source of the information you are looking for, but fall slightly short of the mark.
ASME B&PV Code is composed of about 11 Sections, each dealing with some particular aspect of pressure vessel construction. There are also several pressure piping code sections, each dealing with a particular piping service.
You can go to the ASME web site and to their book store to get a brief description of each code section, none of which are stand alone documents. You need to use several code sections to construct a pressure vessel or piping system. None of the code sections are "design manuals" meaning you cannot design and build a vessel using the ASME codes alone. Keep in mind that ASME delineates the bare bones requirements needed to ensure a safe vessel is constructed or a safe piping system is installed. Meeting ASME requirements alone may not satisfy customer expectations.
A brief rundown of the B&PV code sections that may be applicable to your situation includes:
Section II - Part A - Ferrous Metals, Part B - Nonferrous Metals, Part C - Welding Consumables, Part D - Materials Properties
Section V - NDT: qualifications of the inspectors and procedures, specific techniques, but no acceptance criteria.
Section VIII - Unfired Pressure Vessels, Division 1 and Division 2, includes materials of construction, design, fabrication, inspection (w/ criteria) and testing requirements.
Section IX - Welder, Brazer, and Procedure Qualification - acceptance criteria for welder, brazer, and procedure qualification only, nothing to do with acceptance criteria for actual product
Section B31.3 - Process Piping, Several service classes to address requirements of specific applications such as nonflammable nontoxic fluids, flammable fluids, toxic fluids, high pressure piping, etc. each with specific design and construction requirements as well as specific acceptance criteria.
Keep in mind that the requirements of Section IX are subject to modification by the specific construction code (Section 1, Section III, Section VIII, and each piping code). Meeting the requirements of Section IX alone may not meet the requirements of the construction code. For instance, B31.3 for High Pressure Piping does not accept the use of backing rings (unless removed), welder must be qualified by mechanical testing only, notch toughness is a requirement for each material specification (not by P number), etc.
You need to have the owner or the engineer specify what ASME code sections apply to your work. That is not something the inspector should be determining.
Best regards - Al