Let $\mathcal F$ be a (regular) codimension $q$ foliation. A foliated chart is an open set $U$ and a local diffeomorphism $\varphi:U\to R^{q}\times R^{n-q}$, where $\varphi^1=c^1,\dots,\varphi^q=c^q$ are the plaques of that chart. Every foliation has a foliated atlas. Likewise one may define foliations via a maximal foliated atlas.
I wonder whether there is a local description of singular foliations? Are there local coordinates $\varphi$ such that $\varphi^1=c^1,\dots, \varphi^{q'}=c^{q'}$ for $q'\geq q$.
Or is it modelled in terms of tubular neighbourhoods?