It sounds like he makes you feel like giving in (this is not the universal response, but probably the most common one!). If he has discovered that _works_, you have taught him to do it again, and you will have to (fight all your instinct and) teach him otherwise.
At one, putting the kid away someplace felt like punishment (to both of us) because I was often angry/frustrated--and punishment changes the whole equation and adds emotional components that are damaging. If the baby wants something, disengaging (continue calmly with your already established activity, or if you can't be calm, leaving the room and start another non-request-related activity somewhere else) will teach him that screaming "doesn't work" (to get what he wants, which, for a baby, is NEVER "to be ignored' ;) ).
I had screamers who just thought it was fun to scream--I mean, what a turn-on, right? Look at this HUGE thing they can do! And how much MORE fun if it's not only loud, but the people around them freak out! (Sometimes hitting and biting are that innocent too--just "WOW! Check it OUT!") For that, quick consequences (which despite his young age you should tell him in advance, he'll catch some of the words and learn to recognize the reminder and logically associate (x) with (y)) often work: put him down if you are carrying him, or immediately move him to another room/his crib, or you simply "shut down" to him but stay in the same room--all of these remove the "I'm impacting people!" part of the "WOW!" For these to work, you have to not do them in anger, all of them--otherwise the situation becomes about affecting Mommy again.
And when (not if!) he does it at the store? I have finished shopping with my hand over a kid's mouth, them still screaming (be careful not to get bit, and be ready to be very slobbered). Mostly during the screaming stage I didn't take them shopping/out. But reinforcing that no "good" reactions and only annoying or boring reactions happen when they scream mostly defuses that, plus in public there are so many other things and people to see! I think that helps.
Now, when there are two or more kids, they can ramp eachother up ... and then separating (the trigger from the screamer) becomes more obvious. In your case, you just need to figure out if the trigger is you or not!
oh, and--sometimes violence from kids (hitting, biting, screaming) is a reaction to an event. In my kids' case, two of them get violent when they need to pee. (No joke.) So you could watch for that type of trigger too!