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.