Shanghai is a fairly safe city.  Areas of greatest concern are the foreigner targetted nightclub areas such as Mao Ming Road and Ju Le Road.  These areas can be a bit more risky late at night when the establishments close around 2AM.  Right around closing, police cars can be seen guarding the end of the street, so being aware of your surroundings should be enough to stay safe. Keeping to well lit areas late at night is a great way to stay in safe areas.

The bigger issue in Shanghai is pickpocketing.  When walking around pedestrian boulevards and outdoor shopping areas no pocket and no bag is safe.  Again, being aware of your surroundings is the best policy.  Police and security are fairly present in the city in navy uniforms, and the police can be reached by dialing 110.  The pedestrian area of Nanjing Road is the area where travellers should exercise the most caution, especially on the edges by the intersection with Henan Road, and the intersection with Xizang (Tibet) Road.

 When crossing streets, be aware that motorists, bicyclists, and motorcyclists will most likely not stop for you, but try to find the shortest way around you.  Avoid making direct eye contact with motorists, because if you do, he assumes you will stop for him. 

In rainy weather, many public places become extremely slippery as the surfaces are covered in highly polished marble which offer no traction for most types of footwear.

Scams 

You may be approched by beggars or see beggars with small children on the street. These are sometimes run by large underground networks that take kids from villages, then put them on the street to beg. Sometimes these underground operations even physically deform a child in the hope of generating more sympathy money. So don't feed this negative cycle by giving them money. 

You might also be approached by beggars in outdoor dinning areas on Nanjing Road (for example, outdoor bars, coffee shops, cake shops etc), often these are young children who are not really poor but is just looking for an easy way to make money. These kids will often beg by standing near your table, then getting down on their knees to beg, and as a last resort, they will ask you to give them your food instead of money. The quickest way to get rid of these kids is to inform the restaurant staff, or to tell them you will call the police if they don't leave you alone.

In heavily touristed shopping districts such as Nanjing Rd., unattached foreign men may be approached by young ladies inviting them to a special private bar for drinks and conversation.  These are often scams where upon leaving, the victim is presented with an outrageous bill and won't be allowed to leave unless he pays up. 

Arriving at either of Shanghai's airports, you may be approached by someone in the meet and greet area offering taxi services.  These are unlicensed, illegal taxi's and more often than not you will be forced to pay a much higher fare, if not robbed outright.  Always go outside the terminal to an officially marked taxi station.  Avoid any taxi with a license plate beginning with the letter 'X' - those are illegal private taxis.