Usually the federal DOJ or the city or state's government will have crime rate statistics for most cities/counties. Of particular interest to you would be the murder rate per 100,000 people. If a city or county has had 100 murders in the last year, it may sound high until you know that those figures are for a city of 10 million people, giving a murder rate of 1/100,000 or .001% (a very safe rating). By contrast, a city of 10,000 people with 100 murders has a muder rate of 1,000/100,000, or 1% (a very unsafe rating).
As far as breakdowns by neighborhood, I imagine the most granular level is by police precinct or zone, and you'd have to contact said precinct to see if they can give you a murder count and the population. Another measure is officers per 10,000 people in a precinct. The higher that number, the higher the likelihood that officers are needed in that precinct to keep crime under control. However, it's not foolproof; a low number of officers could be the cause of a high crime rate due to underfunding.
I think the best measure of the safety of a particular street corner is its proximity to a police department building or jail. No criminal in their right mind breaks into a house just across the street from the police department. If your apartment is within a few blocks of the local precinct, the crime rate for said apartment is going to be very low. This is generally very easy to find out; Google Earth has a pretty good map of police and fire stations in most U.S. cities, and you simply have to plot the location of said stations to each candidate address. If your "local" fire station must get on a major highway to get to you, you should probably look elsewhere. Same with the local police force, though it's expected that a policeman on patrol will get to you more quickly than a fire truck in the bay; firemen don't go looking for fires.
