Ditto on the annoyance at mothers simply ignoring tantruming babies. HOWEVER, one thing I cannot STAND to see is a mother who will silence a tantruming child at any cost.
I mean, I think that how we respond to a tantrum can take a natural response of the child's (unorganized nervous system, lack of coping strategies, etc.) and turn it into a very learned pattern. That is, as far as I know, the only thing that will prematurely end a true tantrum is bargaining with a child, or giving the child what it wants, when it wants it. And that, I'm afraid, is when tantrums become the province of will, rather than the involuntary, utter lack of will due to young age.
In my experience (being a full-time nanny for many years, not a mom for 6 more months, I admit it) trying to intervene too much is fruitless. I think it is a really good idea to say, "I know you're frustrated. I know, you're very upset. I understand, you would like to get down. That's just not possible now." And then to some extent, along with the acknowlegement, letting the kid go. THAT kind of "ignoring" it isn't being neglectful, it is a matter of not starting a cycle.
It seems to me that when a child screetches and whines in a store and you cut short your errand and take her outside to play, you are letting her know exactly what needs to be done for her to get what she wants. Sure, at first she won't make the connection, because she isn't doing on purpose -- at first. But babies/toddlers are pretty darn efficient at getting their needs met, and I think they catch on, quick.
As for being embarrassed, screw that. I say do your best to relate to your child in a developmentally-appropriate way, and that may mean he needs to express anger and being upset at not getting his way every hour of the day. It's an important lesson! It's equally important to be able to feel okay about being upset and angry about things, sometimes, even at 2.
Personally, once outside, I would still hold the child, but while sitting, in a relaxed way, until the child calmed down. I would say "There are times when Mommy needs to hold you, and I am not doing it to make you mad. I'm sorry you don't like it. I love you, but you can't play until you can calm down with me." If hair pulling was going on, I would put the child in my lap (his back to my tummy) with my arms crossed in front of his body, holding his hands only strongly enough to keep him from breaking away. Again, with as relaxed a body posture as I could manage -- adding your anger won't help. I've done this through several tantrums, but because of hitting behavior. It gave the boy a secure, safe place to be upset, let yet him know physically what was unacceptable (words are a little lost on an upset toddler).
I guess some people would find this too authoritarian, too "because I said so." I prefer to think of it as setting loving limits. To each his own ...