Different regulatory bodies have different scale certification requirements for different scales. The requirements for certification can vary depending on the size of the scale and what is being measured. For example postal scales, scales that weigh small items and scales that measure large quantities of grain all have different requirements.
At the federal level, the National Institute of Standards and Technology (NIST) covers its requirements for scale accuracy in NIST Handbook 44 Section 2 2.20. In most cases, scales are required to weigh values using only a single unit of measure and are required to maintain a zero indication within 1/4 to 1/2 of the scale's unit of measure.
State and Local
Although there is some variation in state and local requirements, most of them follow NIST requirements and so most inspectors, even at the federal level, accept a state or local certification as proof of a scale's accuracy. State and local requirements vary depending on the size and type of scale, but state governments generally offer, and require, some type of scale certification.
The NIST available certificates include a Statement of Accuracy which shows that a scale that has been tested within NIST guidelines, a Statement of Accuracy with Serial Number which includes the serial number of the scale in question, a Traceable Certificate includes everything listed previously, plus a statement of the actual weight and uncertainties and tolerances and an NVLAP Certificate which conforms to American National Standards Institute/National Conference of State Legislatures (ANSI/NCSL) Z540-1 and is required by the military.