After beating the story I decided to go back and clean up all the street crime side missions I'd neglected. I notice that each desk Traffic, Homicide, Vice, & Arson each have their own set of street crimes. I went through each desk in Free Roam mode and completed them all. Yet somehow I'm only at 36/40. Is there something I'm missing? Do you have to complete every case in the story to a minimum number of stars in order to unlock all the side missions? I one-starred one of the cases which is why I ask. Thought maybe you have to get a minimum of three or something. I don't know, I'm out of ideas. If you're in Free Roam shouldn't all unplayed Street Crimes appear as red icons?

I've seen some street crimes that have characters from some cases in them so maybe it it that. Not sure though.

I got one star on a bunch of cases and I got all 40 of them. Occasionally when doing them in free roam it would make me do one I'd already done before which wouldnt count towards the 40.

I got one star on a bunch of cases and I got all 40 of them. Occasionally when doing them in free roam it would make me do one I'd already done before which wouldnt count towards the 40.

Yeah, eventually it would just trigger new ones until it gave me nothing but new ones on that desk.

I've also heard that some street crime missions only appear at certain times of day...not sure if that's actually true or not.

Yeah, the Rockstar Social Club checklist told me specifically which desks still had outstanding street crimes.  If you go into Free Roam for a certain desk you should get a message if you've completed all the crimes for that desk.  It seems a little twitchy - I was missing 3 crimes on Traffic desk but there were only 2 red icons.  I completed those 2 and reloaded Free Roam and suddenly the third one showed up.

Yeah, the Rockstar Social Club checklist told me specifically which desks still had outstanding street crimes.  If you go into Free Roam for a certain desk you should get a message if you've completed all the crimes for that desk.  It seems a little twitchy - I was missing 3 crimes on Traffic desk but there were only 2 red icons.  I completed those 2 and reloaded Free Roam and suddenly the third one showed up.

I would like to do all the side cases too, now I've completed the main story. It's rather strange it gives you side cases you've already done and makes it more time consuming.
 
I wish they implemented a better system for them.

That happened to me, if you go to a desk that isn't completed yet which they will tell you, just go in it again. You will see a red mark for a street crime. After you finish that, quit and go back to it and you'll get more missions that way. Or wait a while until they let you know the next street crime. But still, figuring out where you didn't finish the street crime, look at your map it should be there, quit and go back to it.

To answer both questions:
 
Certain Street Crimes DO only appear at certain times of day, just hang around or just run over shit while you wait for time to pass.
 
ALSO,  each desk has specific Street Crimes, if your not sure if you have them all for a desk get in ANY cop car in that free roam and if you have them all completed for that particular desk it will tell you so in the top left of the screen.