Given a cubic hypersurface $X\subset \mathbb{P}^4$, we study the existence of
Pfaffian representations of $X$, namely of $6\times 6$ skew-symmetric matrices
of linear forms $M$ such that $X$ is defined by the equation $Pf(M)=0$. It was
known that such a matrix always exists whenever $X$ is smooth. Here we prove
that the same holds whenever $X$ is singular, hence that every cubic threefold
is Pfaffian