  For a Frobenius abelian category $\mathcal{A}$, we show that the category
${\rm Mon}(\mathcal{A})$ of monomorphisms in $\mathcal{A}$ is a Frobenius exact
category; the associated stable category $\underline{\rm Mon}(\mathcal{A})$
modulo projective objects is called the stable monomorphism category of
$\mathcal{A}$. We show that a tilting object in the stable category
$\underline{\mathcal{A}}$ of $\mathcal{A}$ modulo projective objects induces
naturally a tilting object in $\underline{{\rm Mon}}(\mathcal{A})$. We show
that if $\mathcal{A}$ is the category of (graded) modules over a (graded)
self-injective algebra $A$, then the stable monomorphism category is triangle
equivalent to the (graded) singularity category of the (graded) $2\times 2$
upper triangular matrix algebra $T_2(A)$. As an application, we give two
characterizations to the stable category of Ringel-Schmidmeier (\cite{RS3}).
