'I have seen other kids throwing major, embarassing tantrums in supermarkets and would be mortified if mine did that.'
I used to think the same thing tbh, but this last year DS has been a nightmare, hes' got worse, for the most part he's a good little boy and tantrums are usually in the house, but it's inevitable he has some out - this is usually because i've said no to something.
so if he lays on the floor (his silent tantrums) then I stand aside and leave him to it, (usually getting knowing looks from other parents and tuts from battleaxes old people) - but DS is in no trouble there, and also I know he's after a reaction as he looks up at me to see if i'm watching him!
if it's a noisey tantrum (as in a shop one day) i've been known to leave everything and just walk out of the shop - was the £1 shop and he wanted a toy I said no to.
Usually thou I think prevention is better to an extent - ie then DS starts to get roudy, I usually say 'oh DS mummy much prefers to hear your lovely singing voice, shall we sing?' and proceed to sing something in a stupid mannor so he's quiet and laughing, (I get odd looks but hey ho!) or I also say, quite voices indoors please DS and then get a gigle and whispers.
as others have said don't reward bad behaviour - i'm of the view to an extent that any reaction to it is what they're looking for so don't give it.
I do think to have a tantrum in one place then wait to get home to sort it properly isn't really fair on the child - every thing with me is done quickly and forgotten about, he'll have his hissy fit, i'll reprimand etc, and then we have a cuddles - ie mummys sorry for shouting (if I shouted) or DS is sorry for getting upset, cuddle then we go off again and he's laughing.