  We show that the canonical random-cluster measure associated to isoradial
graphs is critical for all $q \geq 1$. Additionally, we prove that the phase
transition of the model is of the same type on all isoradial graphs: continuous
for $1 \leq q \leq 4$ and discontinuous for $q > 4$. For $1 \leq q \leq 4$, the
arm exponents (assuming their existence) are shown to be the same for all
isoradial graphs. In particular, these properties also hold on the triangular
and hexagonal lattices. Our results also include the limiting case of quantum
random-cluster models in $1+1$ dimensions.
