Personally, taking dogs shopping would be an absolute nightmare. My two little dogs would bark at everything that moved, they would wrestle with each other for awhile and my boy dog would then find a nice clothing rack to pee on. Nope, not for us. But there are many, many people who take their well-behaved dogs for a public outing and it works very well.
Some large retail chains allow pets, like Home Depot, Nordstrom, Walgreens, Apple and Lowes. As long as your buddy is well-mannered and on a leash, he or she is welcome.
So, if your dog does not enjoy strangers petting his head and children rushing up toward him, then a shopping experience might not be a good plan. Your dog also may not be a shopping candidate if he is not calm around a lot of strangers, or if he pulls on the leash.
Another good option for taking your dog shopping is to let him ride in a pet stroller. Pet Strollers create a safe and protected space for your pet. He or she can see out of the windows, but it generally keeps others from randomly petting your buddy when you are trying to get some things done.
PerfectPetStroller.com carries a wide variety of pet strollers and pet bike trailers