For this lovely little silver trophy, you will need to scratch in all 12 chapters of Stray. Some chapters will force you to scratch to make progress in the story, while other chapters may only have a singular spot where you can use your claws to shred.
Some great news is that this doesn't need to be done in a single playthrough, so, if you miss/missed any of these on your initial playthrough, just load up the chapters you didn't scratch in and the trophy will unlock once you scratch in the final chapter you needed a scratch in.
Chapter 1: Inside the WallsStray 'Territory' trophy guide: Inside the Wall
As you progress through the first chapter, learning the ropes and interacting with your feline family, you will come across a tree about halfway through the level. Simply approach and scratch the tree for a moment.
Chapter 2: Dead CityStray 'Territory' trophy guide: Dead City
After you push the paint can through the glass window, you will drop down into someone's flat. There are two rugs in this room, simply scratch either of these rugs for your progress to be tracked.
Chapter 3: The FlatStray 'Territory' trophy guide: The Flat
In the room where you need to type on the computer to unlock the secret door, there is a rug on the floor in this area that you can scratch.
Chapter 4: The SlumsStray 'Territory' trophy guide: The Slums
Straight after the cutscene ends, where The Guardian alerts the robot citizens that you are not a threat, simply move forward into the room directly in your line of sight, there is a rug on the floor here that you can scratch.
Chapter 5: RooftopsStray 'Territory' trophy guide: Rooftops
When you see the chapter title pop you should be standing on a small roof, on this roof is a rug you can scratch. Make sure you grab this as from my memory there are very few, if any, other places to scratch.
Chapter 6: The Slums Part 2Stray 'Territory' trophy guide: The Slums Part 2
As soon as the chapter starts, simply move forward into Momo's flat and there is a rug you can scratch in front of you.
Chapter 7: Dead EndStray 'Territory' trophy guide: Dead End
After making it into The Doc's apartment, you will go down some stairs, to the right of these stairs is a room with a wall you can scratch.
Chapter 8: The SewersStray 'Territory' trophy guide: The Sewers
The Sewers is the easiest one to miss as there is only a singular spot where you can scratch. Right after you open the big metal door, where you use B-12 to hack the two keypads, go through the door and up the stairs. After reaching the top of the stairs, to the left, there should be a bit of cardboard you can scratch.
Chapter 9: AntvillageStray 'Territory' trophy guide: Antvillage
There are multiple spots you can scratch here. If you follow the ground floor path around you will come to a robot sitting on a sofa, you can scratch the arm of this sofa.
Chapter 10: MidtownStray 'Territory' trophy guide: Midtown
Midtown is a chapter where you will be forced to scratch at some point to progress the story. Just to be safe we have provided an easy spot where you can scratch just to get it out of the way. In the clothing shop — where you will need to steal from later on — there is a rug near one of the entrances that you can scratch.
Chapter 11: Jail
Stray 'Territory' trophy guide: Jail
Chapter 11 is also an area where you will be forced to scratch. You will need to scratch at some keys that are dangling in a car door. For anyone that would like a proper scratch, there is a tree close to where the car is. You will need to make your way past a wired fence to reach it.
Chapter 12: Control RoomStray 'Territory' trophy guide: Control Room
You will be forced to scratch at some wires to progress the story. After you do this, the trophy should pop! Congratulations on your silver trophy!