Assuming a comet is traveling on trajectory X and breaks up from gravitational forces while slingshotting around the Sun, it is mathematically impossible for all remnants to continue to travel on trajectory X. Furthermore, because the comet was broken up by gravitational forces, it is highly likely that these forces will act on each fragment differently, causing a new trajectory for each. This is the “shotgun pattern” you described. Any prediction of the future path of the remnants is impossible without knowing the mass of the remnant, and it’s exact position in the Sun’s field. Therefore
Anyone who tells you they know where the fragments of Comet Ison will go once they leave the Sun’s influence is arrogantly mistaken.
Let’s see about that shall we?
The only pertinent concern for comet ISON is on Jan 14 when we will pass through remnants of its incoming tail. That is the dust which was blown off during its inbound journey and was sufficiently slowed by the solar wind to still be in the vicinity of earth on Jan 14. Ergo, the particles must be small enough for the force of the solar wind to have counter-balanced its incoming speed -which was very high. So anything encountered on Jan 14 will be incredibly small, likely microscopic, on a molecular scale. Anything larger, particularly anything large enough to become a bolide is too large to have been decelerated by the solar wind to still be in our orbital neighborhood.
At its closest approach on Dec 26 comet ISON will be 40 million miles from earth and above the solar plane heading into interstellar space at a high rate of speed.
Based on the enormous initial momentum along with the increase due to the slingshot effect I think its intuitively safe to say the debris will travel along trajectories with a relatively moderate conical divergence from the original one. This is why we have annual meteor showers. The debris tend to stay In a cloud like stream.