The "secret cases" are found in the missions during the investigation. So far I have found 2 investigations that have secret cases:
The secret case is unlocked in the rooms mission, you need to get into storage where you get the Paper Clip tool which unlocks rooms. You can then get into the Employee lounge where a recording is hidden inside the locker, which unlocks the bonus case (and some sweets hidden there too). Found with the help of toucharcade
The secret case is unlocked in the map mission, when revealing the barber shop on the top left corner, the barber will offer a new investigation.
The secret case is unlocked in the rooms mission on the right of the church, on the bulletin board, there is a ,message which opens a new case. Pictured here
The secret case is unlocked in the rooms mission, in the office, inside the middle desk there is a file, this file unlocks the secret case.