The in-service inspection code for your pressure vessel inspection is API Standard 510. The other API and ASME Codes can be used as supplementary information.
The following standards, codes and recommended practices are used for pressure vessel inspection:
API Standard 510; Pressure Vessel Inspection Code: In-Service Inspection, Rating, Repair, and Alteration
API Recommended Practice 572; Inspection Practices for Pressure Vessels
API Recommended Practice 576; Inspection of Pressure-relieving Devices
API Recommended Practice 577; Welding Inspection and Metallurgy
API Recommended Practice 571; Damage Mechanism Affecting Fixed Equipment in the Refining Industry
ASME PCC-2; Repair of Pressure Equipment and Piping
National Board Inspection Code NB-23
ASME Section VIII; Rules for Constructing Pressure Vessels
Section IX, Welding and Brazing Qualifications
The last two items are specific codes for new construction; however, most of the technical requirements for design, welding, NDE, and materials can be applied to the inspection, rerating, repair, and alteration of in-service pressure vessels.
ASME PCC-2, NB-23 and API 510 can be used for pressure vessel repair.
The NB-23 code covers the rules and guidelines for in-service inspection, repair, and alteration of pressure- retaining items.