  Let $\mathfrak{o}$ be a complete discrete valuation ring with finide residue
field $\mathsf{k}$ of odd characteristic, and let $\mathbf{G}$ be a symplectic
or special orthogonal group scheme over $\mathfrak{o}$. For any
$\ell\in\mathbb{N}$ let $G^\ell$ denote the $\ell$-th principal congruence
subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group
$\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup
$G^{\ell+1}$ for some $\ell$, and if its restriction to
$G^\ell/G^{\ell+1}\simeq \mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of
characters of minimal $\mathbf{G}(\mathsf{k}^{\rm alg})$ stabilizer dimension.
In the present paper we consider the regular characters of such classical
groups over $\mathfrak{o}$, and construct and enumerate all regular characters
of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is
greater than two. As a result, we compute the regular part of their
representation zeta function.
