Which bodies regulate aviation in your country, under what basic laws?

Aviation in the US is regulated primarily by the US Department of Transportation (DoT) and the Federal Aviation Administration (FAA) pursuant to title 14 of the Code of Federal Regulations (FARs), 49 USC (Transportation Code), and the corresponding regulations. Regulation of aviation operations

How is air transport regulated in terms of safety?

The FAA regulates safety of commercial and private air transport. Screening passengers and ensuring onboard security is the responsibility of the Department of Homeland Security’s Transportation Security Administration. The National Transportation Safety Board (NTSB) conducts non-criminal aircraft accident investigations.