  We solve the Wess-Zumino consistency conditions of $\mathcal{N}=1$ off-shell
conformal supergravity in four dimensions and determine the general form of the
superconformal anomalies for arbitrary $a$ and $c$ anomaly coefficients to
leading non trivial order in the gravitino. Besides the well known Weyl and
$R$-symmetry anomalies, we compute explicitly the fermionic $\mathcal{Q}$- and
$\mathcal{S}$-supersymmetry anomalies. In particular, we show that
$\mathcal{Q}$-supersymmetry is anomalous if and only if $R$-symmetry is
anomalous. The $\mathcal{Q}$- and $\mathcal{S}$-supersymmetry anomalies give
rise to an anomalous supersymmetry transformation for the supercurrent on
curved backgrounds admitting Killing spinors, resulting in a deformed rigid
supersymmetry algebra. Our results may have implications for supersymmetric
localization and supersymmetry phenomenology. Analogous results are expected to
hold in dimensions two and six and for other supergravity theories. The present
analysis of the Wess-Zumino consistency conditions reproduces the holographic
result of arxiv:1703.04299 and generalizes it to arbitrary $a$ and $c$ anomaly
coefficients.
