  A monoid is called special if it admits a presentation in which all defining
relations are of the form $w = 1$. Every group is special, but not every monoid
is special. In this article, we describe the language-theoretic properties of
the word problem, in the sense of Duncan & Gilman, for special monoids in terms
of their group of units. We prove that a special monoid has context-free word
problem if and only if its group of units is virtually free, giving a full
generalisation of the Muller-Schupp theorem. This fully answers, for the class
of special monoids, a question posed by Duncan & Gilman in 2004. We describe
the congruence classes of words in a special monoid, and prove that these have
the same language-theoretic properties as the word problem. This answers a
question first posed by Zhang in 1992. As a corollary, we prove that it is
decidable (in polynomial time) whether a special one-relation monoid has
context-free word problem. This completely answers another question from 1992,
also posed by Zhang.
