Bunza, hello.
I think you are doing so well just being consistent. I sounds to me like dd is saying these spiteful things to you as a way of getting attention and pushing boundaries. I really hope this is a passing phase, because I agree with threeangles, some of what you describe, like the throwing cans off supermarket shelves, does sound a bit extreme. But lots of it doesn't.
My sons - 8 and 3 - are not good in shops - I don't have mega high expectations of the 3 year old, but the 8 year old will still be much too loud and disobedient. He was worse when he was 5, however, and now doesn't tantrums as such, so to some extent his bad behaviour was a passing phase.
Long ago I developed a thick skin. If he lay on the floor screaming I ignored him. Often I had to, in order to do the necessary shopping. My son, on the whole, was only too pleased to be escorted out of a shop. He mostly only tantrummed if he found the shop or the shopping trip boring.
I found the promise of a treat at the end of the shopping trip or outing - an ice cream or small toy, worked well on his behaviour.
Also, I think as has been said, it's worth seeing if you can avoid the situation that causes a tantrum - internet shopping, etc.
I think you have three elements to consider - the choice of trip - is this particular trip going to spark off bad behaviour?
What you do on the trip, before a tantrum strikes - is dd getting tired, hungry, bored? do we need a break to keep her on an even keel? Depending on your dd, stepping up the preventative action could work wonders.
And what happens during a tantrum - how you choose to deal with it.
No magic answers, but you are not alone in this.