If in the US, the state that the vessel resides usually has a document called Boiler and Pressure Vessel Rules and Regulations. This is the governing document and it lists all the requirements or codes for design, construction, testing, and inspection within that state. This document will lead you to the ASME code or National Board code where appropriate. The timing and extent of inspection after the initial inspection is often subject to insurance company and corporate reliability rules. It can be as easy as a visual inspection and leak check or as in depth as UT to all welds. I work on a chemical processing facility with numerous jacketed vessels. The only inspection done after the initial construction inspection is a yearly visual inspection for leaks.