We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular
hermitian line bundles of finite height. In particular, the theorem applies to
the log-singular metrics of Burgos-Kramer-K\"uhn. Our theorem is thus suitable
for application to some non-compact Shimura varieties with their bundles of
cusp forms. As an application, we treat the case of Hilbert modular surfaces,
establishing an arithmetic analogue of the classical result expressing the
dimensions of spaces of cusp forms in terms of special values of Dedekind zeta
functions
