  A bar-joint framework $(G,p)$ in $\mathbb{R}^d$ is rigid if the only
edge-length preserving continuous motions of the vertices arise from isometries
of $\mathbb{R}^d$. It is known that, when $(G,p)$ is generic, its rigidity
depends only on the underlying graph $G$, and is determined by the rank of the
edge set of $G$ in the generic $d$-dimensional rigidity matroid
$\mathcal{R}_d$. Complete combinatorial descriptions of the rank function of
this matroid are known when $d=1,2$, and imply that all circuits in
$\mathcal{R}_d$ are generically rigid in $\mathbb{R}^d$ when $d=1,2$.
Determining the rank function of $\mathcal{R}_d$ is a long standing open
problem when $d\geq 3$, and the existence of non-rigid circuits in
$\mathcal{R}_d$ for $d\geq 3$ is a major contributing factor to why this
problem is so difficult. We begin a study of non-rigid circuits by
characterising the non-rigid circuits in $\mathcal{R}_d$ which have at most
$d+6$ vertices.
