  We consider the enumeration of walks on the non-negative lattice
$\mathbb{N}^d$, with steps defined by a set $\mathcal{S} \subset \{-1, 0, 1\}^d
\setminus \{\mathbf{0}\}$. Previous work in this area has established
asymptotics for the number of walks in certain families of models by applying
the techniques of analytic combinatorics in several variables (ACSV), where one
encodes the generating function of a lattice path model as the diagonal of a
multivariate rational function. Melczer and Mishna obtained asymptotics when
the set of steps $\mathcal{S}$ is symmetric over every axis; in this setting
one can always apply the methods of ACSV to a multivariate rational function
whose whose set of singularities is a smooth manifold (the simplest case). Here
we go further, providing asymptotics for models with generating functions that
must be encoded by multivariate rational functions with non-smooth singular
sets. In the process, our analysis connects past work to deeper structural
results in the theory of analytic combinatorics in several variables. One
application is a closed form for asymptotics of models defined by step sets
which are symmetric over all but one axis. As a special case, we apply our
results when $d=2$ to give a rigorous proof of asymptotics conjectured by
Bostan and Kauers; asymptotics for walks returning to boundary axes and the
origin are also given.
