Even in places that are normally low-crime, crime will increase at the holidays because one, there are so many more victims to chose from and two, it's easier to disappear into a crowd. Even if your mall has always seemed safe, it may not be so this time of year. There are often a rash of purse snatchings at malls during this time of year. Where I used to live, men would dive up slowly in a car with the window down, lean out, grab a woman's purse from her shoulder and then floor it out of there. Unfortunately one woman's purse that they grabbed didn't slip off her shoulder, but rather hooked on her arm and they ended up dragging her for many yards before giving up and letting go of her purse.
Always have a purse with a strap (if you don't, you'll be prone to setting it down while you shop and it will become easy bait) and always wear it across your body (bandolier-style). If you are in a store where you use a shopping cart, either keep your purse on you or put it in the bottom of the buggy. Purses stashed in the child seat are easy to lift out. Obviously, never walk away from a buggy that has your purse in it, and if you are shopping with a partner (namely a man) make sure that he is one, aware that your purse is in the buggy and two, that he can't leave the buggy. Most women learn this from their mothers, but men aren't used to toting around something valuable. If he wants to go off, he needs to physically transfer the buggy back to your immediate possession; leaving it behind you while you scour shelves is not acceptable.
And one last thing: I know a lot of women (and even some men) who keep a handgun in their car for protection purposes. Laws vary by state, but in most cases, that is illegal. In TN you have to have a valid handgun carry permit on your person anytime you leave the boundary of your residence/property with a handgun--including when you are in your car. My fiance advocates that it's better to be carried by twelve than carried by six (twelve jurors versus six pall bearers), but at least know the risk you take if you chose to keep a gun in your car. Police do not need a warrant or even a good excuse to search your car. I always thought that the same rules applied to your car as to your house, but that is not the case.