  General $\mathcal{N}=(1,0)$ supergravity-matter systems in six dimensions may
be described using one of the two fully fledged superspace formulations for
conformal supergravity: (i) $\mathsf{SU}(2)$ superspace; and (ii) conformal
superspace. With motivation to develop rigid supersymmetric field theories in
curved space, this paper is devoted to the study of the geometric symmetries of
supergravity backgrounds. In particular, we introduce the notion of a conformal
Killing spinor superfield $\epsilon^\alpha$, which proves to generate extended
superconformal transformations. Among its cousins are the conformal Killing
vector $\xi^a$ and tensor $\zeta^{a(n)}$ superfields. The former parametrise
conformal isometries of supergravity backgrounds, which in turn yield
symmetries of every superconformal field theory. Meanwhile, the conformal
Killing tensors of a given background are associated with higher symmetries of
the hypermultiplet. By studying the higher symmetries of a non-conformal vector
multiplet we introduce the concept of a Killing tensor superfield. We also
analyse the problem of computing higher symmetries for the conformal
d'Alembertian in curved space and demonstrate that, beyond the first-order
case, these operators are defined only on conformally flat backgrounds.
