  We study the structure of divergences and universal terms of the entanglement
and R\'enyi entropies for singular regions. First, we show that for
$(3+1)$-dimensional free conformal field theories (CFTs), entangling regions
emanating from vertices give rise to a universal contribution $S_n^{\rm univ}=
-\frac{1}{8\pi}f_b(n) \int_{\gamma} k^2 \log^2(R/\delta)$, where $\gamma$ is
the curve formed by the intersection of the entangling surface with a unit
sphere centered at the vertex, and $k$ the trace of its extrinsic curvature.
While for circular and elliptic cones this term reproduces the general-CFT
result, it vanishes for polyhedral corners. For those, we argue that the
universal contribution, which is logarithmic, is not controlled by a local
integral, but rather it depends on details of the CFT in a complicated way. We
also study the angle dependence for the entanglement entropy of wedge
singularities in 3+1 dimensions. This is done for general CFTs in the smooth
limit, and using free and holographic CFTs at generic angles. In the latter
case, we show that the wedge contribution is not proportional to the
entanglement entropy of a corner region in the $(2+1)$-dimensional holographic
CFT. Finally, we show that the mutual information of two regions that touch at
a point is not necessarily divergent, as long as the contact is through a
sufficiently sharp corner. Similarly, we provide examples of singular
entangling regions which do not modify the structure of divergences of the
entanglement entropy compared with smooth surfaces.
