Visible crew/equipment: In Wesley's quarters, when the camera looks at Geordi, there appears to either be a person reflected on Wesley's device or a boom mic shadow plays across it.
Visible crew/equipment: When Dr. Crusher is trying to save the Borg, just before Worf says "lower the force field" a boom mic is visible.
Visible crew/equipment: Numerous times throughout the series, the screws that hold on the panel faces on the Conn and Ops positions are visible.
When The Bough Breaks - S1-E17
Visible crew/equipment: When Data and Geordi come up to address Picard, there are two lines of tape on the floor for positions.
Visible crew/equipment: As Picard goes to sit in his seat, there's a shadow of the boom mike playing across the wall above Burke and Kolrami's heads.
Bloodlines - S7-E22
Visible crew/equipment: When Picard and his Jason have their "father and son" chat in the caves, you can see the mike shadow moving up and down a few times on Jason's leg. (00:33:20)
Visible crew/equipment: In the Medical center, as the camera pulls back from Picard, who is lying on the bio bed, the camera is reflected on the panels angled toward the camera.
Encounter at Farpoint (1) - S1-E1
Visible crew/equipment: As Zorn leans over his desk after Riker leaves, the studio light is reflected on the monitor screen to his left.
Where Silence Has Lease - S2-E2
Visible crew/equipment: As Picard approaches Data's position, the boom mic is reflected on the panel behind him.
Visible crew/equipment: As Kolrami talks to Picard from beside the aquarium, you can see the studio light panels reflected on the glass.

Visible crew/equipment: When Worf addresses Klingon captain K'Nera, a microphone dips into the shot over Worf's head. (00:33:10)
Visible crew/equipment: In the scene in the cavern where Wesley fires his phaser to heat the rocks to keep him and Picard warm, Wesley's shadow is in the wrong place. The camera is in almost a direct line of sight between where the rocks are (which are supposed to be providing the light source) and Wesley with the cavern wall behind him. After Wesley fires his phaser and the room illuminates, his shadow should be either directly behind him or to the left of him, but his shadow appears to the right side of him. The only way his shadow would be there was if the light source was off camera a good distance to the left.
All Good Things... (2) - S7-E26
Visible crew/equipment: Captain Picard joins the poker game and Data asks the Captain if he wants to deal. Picard says "Thank you Mr. Data." Riker sits down, and just before Captain Picard starts saying "Actually..." a camera or other bit of equipment briefly pops into shot behind Riker.
Suggested correction: That's not equipment; that's Geordi's VISOR briefly popping in. You can see Geordi is seated next to Riker in other shots.