If you are in Free Roam (accessible from "Cases" via the main menu), the street crimes for the desk you are free roaming under will be displayed on your map (your full map from the pause menu is the best way to find them) if they are available at that particular time (some Street Crimes are only available at certain times of day). If you haven't completed the Street Crime, the icon on the map will be red. If you have completed it, it will be grayish. If you have responded to the radio dispatch concerning the Street Crime, the icon will be yellow.
You can also see how many Street Crimes you have completed by looking at the Rockstar Social Club Stats from the main menu (I think it's "Y" on Xbox), or by looking at your stats on the Rockstar Social Club website.
For information on completing Street Crimes in L.A. Noire, look at this post.
