There have certainly been reports here of strollers being stolen, so if you have a very expensive model, you might want to look into some kind of stroller lock. Just be sure that the CM's will be able to move your stroller even with the lock on or they'll cut it off.
We're way past the stroller stage so perhaps someone else will chime in with better advice on that.
As for the shops, they are often cramped and crowded so a double-wide stroller would not only be tough to maneuver, but highly annoying to other shoppers.