  In this paper, we prove that the algebra of an \'etale groupoid with totally
disconnected unit space has a simple algebra over a field if and only if the
groupoid is minimal and effective and the only function of the algebra that
vanishes on every open subset is the null function. Previous work on the
subject required the groupoid to be also topologically principal in the
non-Hausdorff case, but we do not. Furthermore, we provide the first examples
of minimal and effective but not topologically principal \'etale groupoids with
totally disconnected unit spaces. Our examples come from self-similar group
actions of uncountable groups. More generally, we show that the essential
algebra of an \'etale groupoid (the quotient by the ideal of functions
vanishing on every open set), is simple if and only if the groupoid is minimal
and topologically free, generalizing to the algebraic setting a recent result
for essential $C^*$-algebras.
  The main application of our work is to provide a description of the simple
contracted inverse semigroup algebras, thereby answering a question of Munn
from the seventies.
  Using Galois descent, we show that simplicity of \'etale groupoid and inverse
semigroup algebras depends only on the characteristic of the field and can be
lifted from positive characteristic to characteristic $0$. We also provide
examples of inverse semigroups and \'etale groupoids with simple algebras
outside of a prescribed set of prime characteristics.
