  In this paper we analyze in detail a collection of motivating examples to
consider $b^m$-symplectic forms and folded-type symplectic structures. In
particular, we provide models in Celestial Mechanics for every $b^m$-symplectic
structure. At the end of the paper, we introduce the odd-dimensional analogue
to $b$-symplectic manifolds: $b$-contact manifolds.
