Visible crew/equipment: When Costello is looking at the dead corpses on the beach, two beams of light get reflected on his glasses.
Visible crew/equipment: When the song starts, right before the main title, we have a dolly shot of Billy in jail. It starts from a red wall and moves towards him. If you look closely at the wall, you can see the subtle shadow of the camera and the rest of the equipment as it moves.
Visible crew/equipment: When Queenan is in the subway and looks to see if he has a tail following him, you can see a boom mic appear and disappear in the top right corner.
Visible crew/equipment: When Ellerby introduces Dignam and he describes his undercover unit, a microphone is seen moving around at the top, center of the screen. (00:23:10)
Visible crew/equipment: When Costello shows Billy the mutilated hand in a bag, filming crew members standing up are reflected on the wine glass.
Visible crew/equipment: When Queenan dies after being thrown out of the building and his body is crushed on the ground in front of Costigan, as he looks at his hand covered in blood, in the shot on the top on the left side of the screen the dolly track of the camera is visible for a second.
Answer: The FBI and police are separate organizations that do not share full information. Only the FBI and Costello himself knew about the informing, and the FBI can only do so much to stop Costello being caught without drawing attention to their relationship. The police and Costello's gang are fighting a separate battle, hence the two moles.