  It is known that Hirzebruch surfaces of non zero degree do not admit any
constant scalar curvature K\"ahler metric \cite{ACGT,G,M17}. In this note, we
describe how to construct Hermitian metrics of positive constant Chern scalar
curvature on Hirzebruch surfaces using Page--B\'erard-Bergery's ansatz
\cite{P78,B82}. We also construct the interesting case of Hermitian metrics of
zero Chern scalar curvature on some ruled surfaces. Furthermore, we discuss the
problem of the existence in a conformal class of critical metrics of the total
Chern scalar curvature, studied by Gauduchon in \cite{G80,G84}.
