Dehn invariant zero tetrahedra

We survey literature on all known families and examples of Dehn invariant zero tetrahedra. We also contribute two previously unknown families of Dehn invariant zero tetrahedra. Following a suggestion of Dill and Habegger, we show that there are finitely many Dehn invariant zero tetrahedra whose dihedral angles have a five-dimensional span over $\mathbb{Q}$, a first step towards classifying all Dehn invariant zero tetrahedra.