There is a class of what I call smoothly stratified spaces'. This is a bit less general than the larger class of stratified spaces (satisfying Thom-Mather axioms) in that it doesn't allow cusps. These spaces come up in many settings, but definitely include all polygonal and polyhedral domains. They are the setting for analysis in Cheeger's old paper Spectral geometry of cone-like spaces' JDG 1983 (?), and are also discussed in detail in a recent paper of mine with Albin, Leichtnam and Piazza (The signature package on Witt spaces, just came out in Ann ENS). One point is that if you have a polyhedron, e.g. in $\mathbb{R}^3$, with vertices which are more than trivalent, then it is simply not a manifold with corners, but you can think of the local structure near each vertex as a cone over a spherical polygon.

There is a class of what I call "smoothly stratified spaces". This is a bit less general than the larger class of stratified spaces (satisfying Thom-Mather axioms) in that it doesn't allow cusps. These spaces come up in many settings, but definitely include all polygonal and polyhedral domains. They are the setting for analysis in Cheeger's old paper `Spectral geometry of cone-like spaces' JDG 1983 (?), and are also discussed in detail in a recent paper of mine with Albin, Leichtnam and Piazza (The signature package on Witt spaces, just came out in Ann ENS). One point is that if you have a polyhedron, e.g. in $\mathbb{R}^3$, with vertices which are more than trivalent, then it is simply not a manifold with corners, but you can think of the local structure near each vertex as a cone over a spherical polygon.