  In four spacetime dimensions, all ${\cal N} =1$ supergravity-matter systems
can be formulated in the so-called $\mathsf{U}(1)$ superspace proposed by Howe
in 1981. This paper is devoted to the study of those geometric structures which
characterise a background $\mathsf{U}(1)$ superspace and are important in the
context of supersymmetric field theory in curved space. We introduce
(conformal) Killing tensor superfields $\ell_{(\alpha_1 \dots \alpha_m) ({\dot
\alpha}_1 \dots {\dot \alpha}_n)}$, with $m$ and $n$ non-negative integers,
$m+n>0$, and elaborate on their significance in the following cases: (i)
$m=n=1$; (ii) $m-1=n=0$; and (iii) $m=n>1$. The (conformal) Killing vector
superfields $\ell_{\alpha \dot \alpha}$ generate the (conformal) isometries of
curved superspace, which are symmetries of every (conformal) supersymmetric
field theory. The (conformal) Killing spinor superfields $\ell_{\alpha }$
generate extended (conformal) supersymmetry transformations. The (conformal)
Killing tensor superfields with $m=n>1$ prove to generate all higher symmetries
of the (massless) massive Wess-Zumino operator.
