In 1995 Mukai showed that a general smooth genus 7 curve can be realized as the intersection of the orthogonal Grassmannian OG(5,10) in $\mathbb{P}^{15}$ with a six-dimensional linear subspace, and that the GIT quotient Gr(9,16)//SO(10) is a birational model of $\overline{M}_{7}$. Which singular objects appear on the boundary of Mukai's model? As a first step in this study, calculations in Macaulay2 and Magma are used to find and analyze linear spaces yielding three singular curves: a 7-cuspidal curve, a family of genus 7 graph curves, and the balanced ribbon of genus 7.