Conditions for underlying space of an orbifold $\Bbb T^n/\Gamma$ to be a sphere? I think you could be overlooking one of the conditions of Dunbar when you say 'most' quotients of $T^3$ have underlying space $S^3$. In fact, there are several quotients of $T^3$ with underlying space $S^1\times S^2$, for example the product of the pillowcase $S^2(2,2,2,2)$ and $S^1$. These all turn out to be fibered orbifolds, so Dunbar doesn't focus on them. However, it should be pointed out that $S^2(2,2,2,2) \times S^1$ can cover itself, so 'most' perhaps might be 'most (up to homeomorphism)'.