I know the following example from Kontsevich. Not sure it is suitable to your course, but I like it.

Consider a space of closed 6-edges polygonal lines in Euclidean ${\mathbb R}^3$ such that each edge has length $1$ and each angle between subsequent edges is the right angle. The problem is to count the virtual and real dimension of that space (modulo the action of Euclidean group). Surprisingly, the quotient space has isolated points and component of the dimension $1$.