We present the first detailed, large-scale study of the Minimal Supersymmetric Standard Model (MSSM) at a $sqrt s=500$ GeV International Linear Collider, including full Standard Model backgrounds and detector simulation. We investigate 242 points in the MSSM parameter space, which we term models, that have been shown by Arkani-Hamed et al to be difficult to study at the LHC. In fact, these points in MSSM parameter space correspond to 162 pairs of models which give indistinguishable signatures at the LHC, giving rise to the so-called LHC Inverse Problem. We first determine whether the production of the various SUSY particles is visible above the Standard Model background for each of these parameter space points, and then make a detailed comparison of their various signatures. Assuming an integrated luminosity of 500 fb$^{-1}$, we find that only 82 out of 242 models lead to visible signatures of some kind with a significance $geq 5$ and that only 57(63) out of the 162 model pairs are distinguishable at $5(3)sigma$. Our analysis includes PYTHIA and CompHEP SUSY signal generation, full matrix element SM backgrounds for all $2to 2, 2to 4$, and $2to 6$ processes, ISR and beamstrahlung generated via WHIZARD/GuineaPig, and employs the fast SiD detector simulation org.lcsim.
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