i know that flag ops are international operations for a 121 air carrier. Supplemental "i believe" is for anything other than domestic and flag, and usually entails charter operations that the airline may do. But i'd double check me on the supplemental cuz im not 100% positive. Hope this helps!
Supplemental authorizes a 121 carrier to operate on-demand charter. For example, many regionals fly athletic teams around on their aircraft. Most regionals, nationals, and majors are certified supplemental as well as flag and domestic.