  For stochastic Hamilton-Jacobi (SHJ) equations, instability points are the
space-time locations where two eternal solutions with the same asymptotic
velocity differ. Another crucial structure in such equations is shocks, which
are the space-time locations where the velocity field is discontinuous. In this
work, we provide a detailed analysis of the structure and relationships between
shocks, instability, and competition interfaces in the Brownian last-passage
percolation model, which serves as a prototype of a semi-discrete inviscid
stochastic HJ equation in one space dimension. Among our findings, we show that
the shock trees of the two unstable eternal solutions differ within the
instability region and align outside of it. Furthermore, we demonstrate that
one can reconstruct a skeleton of the instability region from these two shock
trees.
