How can I tell which street crimes I have solved?

I don't know if you can tell from the case selection screen, but when you're in Free Play mode, you can check the map. Street crimes you've solved are shown as gray icons (I believe they look like a person walking) on the map. If all you see is gray icons, you've done all the street crimes for that desk.

If you see one you haven't done, you can set it as your waypoint, and then you'll get that crime immediately - without having to wait for the dispatcher to work its way around to that one. This really helps if there's just one or two crimes left to go on a desk, or if there's a set of them close together that you'd like to tackle as a group.

This guide seems to back me up, and also makes note of a couple of other caveats:

How do you know which ones you have completed?

The ones you've completed show up as grey on the map and the ones you havent will be red. Free roam shows all available street crimes at that time frame [whereas] in-story only shows when [you're] in the vicinity.