  Although the leading-order scaling of entanglement entropy is non-universal
at a quantum critical point (QCP), sub-leading scaling can contain universal
behaviour. Such universal quantities are commonly studied in non-interacting
field theories, however it typically requires numerical calculation to access
them in interacting theories. In this paper, we use large-scale T=0 quantum
Monte Carlo simulations to examine in detail the second R\'enyi entropy of
entangled regions at the QCP in the transverse-field Ising model in 2+1
space-time dimensions -- a fixed point for which there is no exact result for
the scaling of entanglement entropy. We calculate a universal coefficient of a
vertex-induced logarithmic scaling for a polygonal entangled subregion, and
compare the result to interacting and non-interacting theories. We also examine
the shape-dependence of the R\'enyi entropy for finite-size toroidal lattices
divided into two entangled cylinders by smooth boundaries. Remarkably, we find
that the dependence on cylinder length follows a shape-dependent function
calculated previously by Stephan {\it et al.} [New J. Phys., 15, 015004,
(2013)] at the QCP corresponding to the 2+1 dimensional quantum Lifshitz free
scalar field theory. The quality of the fit of our data to this scaling
function, as well as the apparent cutoff-independent coefficient that results,
presents tantalizing evidence that this function may reflect universal
behaviour across these and other very disparate QCPs in 2+1 dimensional
systems.
