I gave some more details below.
But to your specific questions the break in the neutral can be at the trasnformer, the arial or underground cable, where it is spliced to the feeder wires at the weather head, at the connections to the meter, the meter to service disconnect (usually the main panel), or from the serice disconnect to subpanels, if any. Of courese if the problem is feeder to a sub-panel then it only affects those loads that are on the sub-panel.
But the connection to the transformer, and the weahter head are all exposed to weather and are most likely source of problems.
One of the hots could have also failed and they power would be lost to 1/2 of the 120 loads and the 240 loads won't work correctly. Or if turn on something like an electric stove then it might back beed the other 120 v loads, but the stove would barely get warm.
As to a shock hazzard. There really should not be one, ALL OTHER THINGS BEING correct. At the serivce disconnect (most often the main panel) the neutral is bonded to the Ground Electrode wiring (what goes to ground rods, underground metal water pipe), bonds to the water pipe if it is metal inside, but not outside, and the Equipment Grounding Conductors (the EGC or what is common called "the ground").
The whole idea is that inside the house everything be at the same potential. It really does not matter what that is, but if everything in the house is the same potential there is no shock hazard.
But also note that older house sometimes have defective grounding and bonding.
One example of this is that it used to be allowed to attach EGC's to local coldwater pipes. Now if a section of that pipe is replaced with plastic pipe you lose that ground connection. That is why it is no longer allowed.
But, as you found out, there is danager to the equipment in use and that can start a fire.
Also the problem indicates a bad connection and if it is in the panel or meter or internal wiring that bad connection can heat up enough to cause a fire.