  This paper is the first in a series of four papers aiming to describe the
(almost integral) Chow ring of $\bar{\mathcal{M}}_3$, the moduli stack of
stable curves of genus $3$. In this paper, we introduce the moduli stack
$\tilde{\mathcal{M}}_{g,n}^r$ of $n$-pointed $A_r$-stable curves and extend
some classical results about $\bar{\mathcal{M}}_{g,n}$ to
$\tilde{\mathcal{M}}_{g,n}^r$, namely the existence of the contraction
morphism. Moreover, we describe the normalization of the locally closed
substack of $\tilde{\mathcal{M}}_{g,n}^r$ parametrizing curves with
$A_h$-singularities for a fixed $h\leq r$.
