No Laughing Matter
Clue #1: Hand-drawn Blackgate map
When you wake up in the destroyed room in Administration, look to the bench in the foreground. A clue rests there.
Clue #2: Ventilation manual
When you emerge from a vent to fight several Joker thugs, take them out and move left. The clue is on a discarded couch.
Clue #3: Cylinder cap
On your way to the construction area, you'll move through a vent with both Joker gas and electrical nodes. When you emerge from this, use the Batclaw to rip down a vent across from you. Climb through here to find a clue on a bookshelf.
Cell Block Bird Cage
Clue #1: Electrical wiring
After shutting down the dual laser doors in Cell Blocks, head through the left opening. The clue is by a gate.
Clue #2: Broken security camera
When you reach the huge room with floors of cells, Batman will remark that you're close to the exercise yard. The clue is on the bottom right of this room.
Clue #3: Mirror shard
With the Batclaw in your possession, return to the room with the improvised bomb. Above the hole it created is a vent you can rip out of place. Do so, then crawl through to find this just after the vent drop-off.
Clue #1: Bullet casings
After activating the pistons in Industrial, head left, Climb the ledge and use the Crypto Sequencer to open the alpha lock and enter a large room. In the bottom left corner of this room is the case's first clue.
Clue #2: Severed thumb
In the long hall before the Black Mask boss battle, there will be a spot to grapple to the upper level. Do so, then head to your left. Use the Line Launcher to cross the gap, then look for this clue on a box.
Clue #3: Cell phone
On your way to the secret entrance to the warden's office, stop in the room with the four large pipes. Just before the last pipe is a spot on the ceiling you can blow through with the Gel Launcher. Do so, then grapple up here to find a clue atop some boxes.