This Is the Police detective case investigations guide
There are a lot of things to get frustrated about in This Is the Police. Send some officers out to a simple call and have one of them die? Check. City Hall riding your back and cutting funding? Check. Investigations seeming impossible? Double check.
It's not totally mandatory that you finish investigations. Chances are, there are going to be a few you never finish. Your detectives may take an eternity to find all the frames, you may never figure out the order of an investigation's frames, your officers may fail to catch the suspect once it's all said and done. And let's not get into full gang investigations.
The first big issue you run into past the three-frame investigations is your detectives not being able to get all of the frames you need to come up with the solution. This really gets some players, and admittedly sometimes it takes entirely too long to get all the frames even with a few detectives on the case.
One thing to note is you cannot see new investigation opportunities if all your detectives are working cases. If you'd like to get more investigations sometimes it's best to have one detective free just to have new ones appear on the map.
Can't get the frames? Put your detectives to work
If you can't solve an investigation and you think (or know) you need more frames, you need to place more detectives on the case. You can put detectives from both A and B shifts on a case regardless of the shift the case was started on. The more detectives on a case, the better.
Don't forget that your detectives' ability to solve investigations is directly tied to their professionalism, though even detectives with professionalism in the high hundreds can find new frames at a snail's pace. Always remember you can go to your Deputy and send your officers out to the police academy to increase their professionalism, even your detectives.
The investigation solution images below are not all you have to solve in This Is the Police, they are simply the ones I finished and remembered to screenshot. I'm playing through a new game so I will be adding investigations I forgot to include here as I push forward through once again.
[Added 1 additional case August 18]
Arson (3 puzzle frames)
Robbery (3 puzzle frames)
Homicide (3 puzzle frames)
Receiving stolen property (Red Mask Gang investigation) (3 puzzle frames)
Sale of stolen property (Red Mask Gang) (3 puzzle frames)
Hit and run (4 puzzle frames)
Drug possession (Atala Funeral Home Gang) (4 puzzle frames)
Theft (5 puzzle frames)
Theft (5 puzzle frames)
Hit and run (5 puzzle frames)
Drug sales (5 puzzle frames)
Homicide (5 puzzle frames)
Drug trafficking (Atala Funeral Home Gang) (5 puzzle frames)
Homicide (6 puzzle frames)
Drug possession (6 puzzle frames)
Rape (8 puzzle frames)
Double homicide (8 puzzle frames)
Vargas investigation (Vicus Vargas gang) (8 puzzle frames)
As mentioned these are by no means all the investigations you have to solve in This Is the Police, and there are certainly gaps. Missing investigations will be filled out to some extent as I push through my second playthrough. For now this is it, but hopefully these solutions will come in handy.
I've also worked up a guide on the Dentist investigation, starting from hunting down the evidence and ending at the frames and sequence needed to figure out his MO. It's not perfect but it does the job.