Is there a case such that $S$ is a manifold on its own right but not a submanifold?

Question

To be more precise, as sets, $S\subset M$, and S has structure such that it is a manifold on its own right but not a submanifold of the manifold $M$?

Answer 1

Take $M = \mathbb R^2$. Take $S$ to be the image of $i: \mathbb R \to M$ given by $i(t) = (t^2, t^3)$. Then $S$ has a cusp at $(0,0)$, so it does not inherit a manifold structure from $M$. However, $S = \mathbb R$ as a set, which has an obvious manifold structure of its own.