I had a small dog, (16 pound West Highland Terrier), when I was younger.
She was extremely well behaved. And, would sit quietly, in the seat of the shopping cart, while I shopped in HD, or Lowes.
One day, an associate stopped me while entering, and said, "Only assistance animals were allowed".
I said with a straight face, "This dog is an assistance animal, she is a lumber picking dog".
The associate called his manager, and then said, "Okay". As long as she remains in the cart, and doesn't cause any problems.
Unfortunately, most people don't seem to have much control over their animals. Relieving themselves in the store, is a good example of that.