  We show that an elliptic modular form with integral Fourier coefficients in a
number field $K$, for which all but finitely many coefficients are divisible by
a prime ideal $\frak{p}$ of $K$, is a constant modulo $\frak{p}$. A similar
property also holds for Siegel modular forms. Moreover, we define the notion of
mod $\frak{p}$ singular modular forms and discuss some relations between their
weights and the corresponding prime $p$. We discuss some examples of mod
$\frak{p}$ singular modular forms arising from Eisenstein series and from theta
series attached to lattices with automorphisms. Finally, we apply our results
to properties mod $\frak{p}$ of Klingen-Eisenstein series.
