I would use public arrest/crime data tied together with data from users and "Area Contributors", which will be locals paid a small amount to find and report crime across a city. In order to keep the data accurate and troll-proof, it would be cross-checked with police arrest data and crime statistics once entered. Users would also have to enter some kind of proof the event actually happened (A picture, police report, etc). This crime report would be marked as pending and would then be approved or denied by an admin - in order to keep racist, invalid, or generally irrelevant reports out of the database. Note that area contributor reports would be instantly approved. I may remove or add some of the user barriers to entry at a later time.
As for your example, a bike stolen or car broken into would probably be put down as a low-profile report and wouldn't really affect the area's safety rating unless it happened often. I want to ensure that average neighborhoods aren't marked as high-crime simply due to a few petty thefts, however areas prone to theft would absolutely be marked as such.
Remember that the app is still in it's planning stages and still could be cancelled altogether or some features could change.