A G2-manifold with conical singularities is also called a $G_2$-conifold.
On G2-conifolds (G2-manifolds with conical singularities):
Survey:
Spiro Karigiannis, $G_2$-conifolds: A survey, 2014 (pdf)
Mark Haskins, Exotic Einstein metrics on $S^6$ and $S^3 \times S^3$ nearly Kähler 6-manifolds and $G_2$-holonomy cones, 2016 (pdf)
Three simply connected $G_2$-cones are known: the
metric cone on $S^3 \times S^3$
metric cone on $SU(3) / (U(1) \times U(1))$
Robert Bryant, Simon Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. Volume 58, Number 3 (1989), 829-850 (euclid:euclid.dmj/1077307681)
Gary Gibbons, Don Page, Christopher Pope, Einstein metrics on $S^3$, $\mathbb{R}^3$ and $\mathbb{R}^4$ bundles, Comm. Math. Phys. Volume 127, Number 3 (1990), 529-553 (euclid:cmp/1104180218)
More on the metric cone over complex projective 3-space as a G2-manifold:
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