Cross-body bags are best to prevent theft, though to be honest, I've heard thieves can and do cut the strap and run with the purse. I always have a cross-body bag (Fossil, two-tone supple leather and deliriously cute) and would never go back. Also keeps my hands free to hold more important things - like little hands.
I always scan parking lots as well, and I make sure the first thing I do is get my sweet cherubs in the van and shut them in while they buckle up. If there's no cart corral VERY close to me, I leave the cart out of the way nearby. I won't chance leaving those kids for one second. Sometimes women forget that children make one infinitely more vulnerable - threaten my child and I will cooperate instinctively. Bad guys know this. I remember hearing of a murderer who used to frequent parking lots of malls, targeting mothers with young kids, laden with shopping bags. He preferred mothers with SUVs and tinted windows, which suited his purposes. He knew menacing their babies = instant cooperation from Mom. Once inside, he'd zip-tie the mom's neck to the head rest and she was effectively incapacitated.
These people are seriously evil. We cannot let our guards down...