Spider-Man 3

As you are swinging through the city, keep an eye open for yellow blips to appear on your radar. These are crimes. Use your Spidey-sense, and you will be able to clearly see the criminals in red through the walls of nearby buildings. Defeat all the criminals to complete the mission and get the message "Crime Averted." Avert 25 of these crimes for the Achievement.

If Spider-Man says something specific about the crime when you approach the area, it is not a random city mission, and instead falls under the heading of petty theft, bank robbery, etc, for which there are separate Achievements. Random missions are typically police chases, gang brawls, or muggings.

These are easy just find a random crime mission they appear as yellow dots on the mini map and can range from beating up a few gang members to stopping a runaway car it just takes time to complete.

Walk around the city with the spider sensor activated, appears faster when active. And keep an eye on the mini map, because at any minute a yellow dot may appear that means a crime has occurred.