  In this paper we give several conditions for a space to be minimal for
conformal dimension. We show that there are sets of zero length and conformal
dimension 1 thus answering a question of Bishop and Tyson. Another sufficient
condition for minimality is given in terms of a modulus of a system of measures
in the sense of Fuglede \cite{Fug}. It implies in particular that there are
many sets $E\subset\mathbb{R}$ of zero length such that $X\times Y$ is minimal
for conformal dimension for every compact $Y$.
