Police said the incidents have occured in the town’s Pinkie Road, between midnight and 4am, and that enquiries have since established three boys are involved.
A statement released by East Lothian Police on social media said: “They have been observed hiding behind the wall running alongside Pinkie Road on the same side as Somerset Fields.
“Can local residents please check any ring-go or similar CCTV systems and if anyone is seen in the early hours acting in a suspicious manner in this area contact 101 to pass on any information to the Musselburgh Community Action Team.
“Please be vigilant and dial 999.”
Local police said there have been “a number of reported incidents” in recent weeks of bus windows being smashed while the vehicles travelled along Pinkie Road.
In March, Lothian Buses suspended all Edinburgh evening services after months of antisocial behaviour incidents on or around buses across the city.
The trigger for the decision was when eight buses serving the Edinburgh Royal Infirmary were targeted by vandals in Old Dalkeith Road or Gilmerton Road, resulting in the removal of services on these routes.
Police said a number of the buses had large stones thrown at their windows and described the actions as “senseless” that could lead to serious injury or even death.