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We study the codimension two locus $H$ in ${𝒜}_g$ consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class $\left[H\right]\in C{H}^2\left({𝒜}_g\right)$ for every $g$. For $g=4$, this turns out to be the locus of Jacobians with a vanishing theta-null. For $g=5$, via the Prym map we show that $H\subset {𝒜}_5$ has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of $\overline{{𝒜}_5}$ and show that the component $\overline{N_0^{\text{'}}}$ of the Andreotti-Mayer...