Which tree has branches that break the least over all other trees? Which trees shed the least weight?
Shedding 'weight' I assume you mean leaves and branches? That would depend on the size of the tree, the age of the tree, what time of year. Conifers would be my guess for 'shedding less weight' since they do not abscise their leaves for winter dormancy. There are a few conifers that are deciduous, however.
On the subject of branches breaking, the trees that have the widest angle between branch and trunk or more than a 45 degree angle up to 90 degrees have the strongest branches. The tighter the angle between branch and trunk the weaker that connection will be.
Oaks are a good example of this. Almost 90 degrees from the trunk. Those branches are less likely to break. Birch or poplar with very upright branching and angles less than 45 degrees from the trunk (or main branch) have very weak connections and those branches break easily.
Two other trees that are less likely to lose branches:
Sugar maples (Acer saccharum). I would go only with the species here, although there are a couple of decent sugar maple hybrids. I would not recommend any other maple species, and certainly watch out for things like Autumn Blaze maples, which are from two different species and break constantly.
Honey Locust (Gleditsia triacanthos inermis) and hybrids. Note that some specimens of this tree can be brittle however, so even though they have good branching structure, you may still lose smaller branches on occasion.
A note on oaks - if Oak Wilt is in your area, than it's essential that you prune your tree during your winter (December through February in the northern US).
I guess it depends on the context. I lived through the ice storm of January 1998 and speaking from experience what happened was that the hickories were about the worst. It was quite worrying to hear the major branches snapping loudly and frequently. Mostly the main trunks stayed intact, just the laterals breaking. Ash was also quite bad. Many maple syrup operators reported that their trees were damaged and saw a risk of infection in the open wounds but I think they were just hoping for $$ compensation.
Those trees that came through best were conifers (as Stormy notes, upvoted) since they are accustomed to taking weight and bending soft laterals under the weight of snow/ice while the main trunk stays straight. Birches responded by bending like reeds, the whole tree bowed over so the tips touched the ground, and then when the ice melted 10 days later just straightened up and carried on as normal. Birches are however quite dirty trees and will drop lots of little twigs all over the place in normal conditions. Red oak was hardly touched the branching is so reliable.
You might want to look at some of the pics of the ice storm events.