DS1 was horrendous for supermarket tantrums when he was 2. Unfortunately it co-incided with me being pregnant and ending up on crutches. He had multiple food allergies so I couldn't shop online in one supermarket to be able to avoid the trigger.
My most memorable one was at 38wks. We'd gone to the supermarket cafe, purely to get out of the house as one of the few accessible places left of mutual interest. He wanted to go into the shop to look at the toy aisle. I was in too much pain to stagger that far. He exploded into the full thrashing, screaming, polishing the floor routine. The only thing I could do was wait it out. I couldn't even bend down to him anymore. A lovely old couple offered to pick him up and put him on a ride on and even put the money in for him which managed to distract and cheer him enough to a state where he could be safely walked across the carpark.
I can't vow that I have never given in, there were that many, most are just a fuzz. Normally an incentive like going to look at toys as a reward for good behaviour works. Reasoned, polite discussion can work.
There was the time he was utterly desperate to have a small car, and it made it as far as the till... oh how he scoured the shopping bags on getting home. That only worked once. Every other time he was permitted something, it was scruitinised hawk-like until the checkout, through the checkout and to the bags!
I suspect there are SN involved. He tantrumed early (10m) and intensively. They did ease significantly when he was 5, but at 8, he still struggles to react rationally to petty stuff like his DB getting the blue cup first. Once he's going, most reactions make little difference to him and I have other areas of concern.