BBS cases are the equivalent of your sleuth cases in Cyber Sleuth and are available very early on in the game, shortly after the start of chapter 2.
You need to use your PC at the cafe and select the BBS icon to open the folder and inside will be your cases. Select a case you wish to work on and complete its requirements then return to the PC to "turn in" your results.
There are 34 main cases that progress the story so you only need to do an additional 16. I got the trophy in chapter 13 and had done every case that came up.