MANPADs are the biggest threat, as far as I know, because they're portable, relatively cheap, and available. The most common ones are SA-7s and Stingers, both IR seeking.
With the big planes, the engines would be what would be hit. The bigger the plane, the further apart they are, so less likely to damage other engines when they're hit. Most of these planes have good engine-out performance, even at gross.
To shoot one down would take a big coordinated strike on takeoff. It's been tried a few times in Iraq, no luck so far, even w/ twin engine jets.
So sure, it's possible, but there's nothing magical about the 380 that makes it a missile magnet.