All airlines are licensed and regulated by the regulator in their own home country. So, British Airways is regulated by the UK CAA and American Airlines by the US Federal Aviation Administration. This oversight carries on regardless of where in the world a flight is. So, if a British Airways aircraft has an incident in Singapore it will file a report to the UK CAA.
All major airlines hold an Air Operators Certificate (AOC). This is the safety licence granted by the CAA and details all the safety aspects of the airline’s operation. If at any time the CAA is not satisfied that an airline meets the requirements of the AOC it can be withdrawn or amended. Without an AOC an airline cannot operate.
As well as a safety licence airlines based in EU states must also have an operating licence. This sets requirements for the financial state of an airline and also requires that it be majority owned by an EU national or company.
The CAA regularly monitors and inspects UK airlines to ensure they are meeting safety requirements. This ranges from audits to CAA staff who are qualified airline pilots flying with the airline on occasional flights.