  It has been realised that corners in entangling surfaces can induce new
universal contributions to the entanglement entropy and R\'enyi entropy. In
this paper we study universal corner contributions to entanglement negativity
in three- and four-dimensional CFTs using both field theory and holographic
techniques. We focus on the quantity $\chi$ defined by the ratio of the
universal part of the entanglement negativity over that of the entanglement
entropy, which may characterise the amount of distillable entanglement. We find
that for most of the examples $\chi$ takes bigger values for singular
entangling regions, which may suggest increase in distillable entanglement.
However, there also exist counterexamples where distillable entanglement
decreases for singular surfaces. We also explore the behaviour of $\chi$ as the
coupling varies and observe that for singular entangling surfaces, the amount
of distillable entanglement is mostly largest for free theories, while
counterexample exists for free Dirac fermion in three dimensions. For
holographic CFTs described by higher derivative gravity, $\chi$ may increase or
decrease, depending on the sign of the relevant parameters. Our results may
reveal a more profound connection between geometry and distillable
entanglement.
