Visible crew/equipment: While Annie, Miss Hannigan, and the inspector are in the bathroom, just as Annie says, "Bye, Mr Inspector," right above Miss Hannigan's hand we can see in the mirror's reflection that something dips into view, presumably the boom mic.
Visible crew/equipment: When Harry and Lloyd first arrive at the Ken Convention in El Paso, there is a shot of the building. Lloyd walks into the building, opening the door. In the reflection, a large light can be seen, as can some of the crew.
Visible crew/equipment: When Kate and Carly are drinking in the window seat of Carly's apartment, Carly goes to to the kitchen to get some vodka. When she's walking back to the window seat, you can see her mike pack showing through the back of her shirt.
Visible crew/equipment: When Eggsy is in the car yelling at his mum's boyfriend and the windows start to go up, you can see a man's head next him on the floor.
Visible crew/equipment: In the scene after Petyr's death, Deacon is standing in front of a table and the window directly above the table is shown. At that point, you can see a boom mike being yanked out of sight. I know this is supposed to be a docudrama, but this is the only place equipment is shown in the entire movie.
Suggested correction: You said it yourself, it's the boom mic of the documentary crew. They make no attempts to hide the fact that it's a documentary.
Visible crew/equipment: When the 2 are walking along side their car at the very start, you can see a driver's head as they go past the building. (00:02:15)
Visible crew/equipment: You can see the camera man in the scene where Megan and her hook up are eating pizza. The window behind them shows the reflection.
Visible crew/equipment: After Riggan ran across Times Square in his underwear and comes into the theatre's entrance, the shot suddenly turns left towards the ticket office, on whose golden shelf the cameraman's legs are reflected. (01:15:04)