Police said passengers on the bus were left "distressed" following the incident in King Street on Sunday evening.
Officers said a group of local youths had also caused "extensive and unnecessary" damage to Greenfield House and cars in Hill Street on Sunday.
A police spokesman said: "This sort of behaviour will not be tolerated."
He added: "Inquiries are currently ongoing to identify those responsible."
A bus passenger was left covered in glass after the vehicle's window was smashed in Alloa at the weekend.