In Applications of Lie Groups to Differential Equations, by Peter J. Olver, the orbit of a Lie group acting on a manifold is defined to be a minimal subset of the manifold that is closed under the group action (i.e. an orbit is closed under the group action and contains no non-trivial proper subsets that are closed under the group action). The orbits of a given Lie group action on a manifold turn out to be submanifolds of the manifold. If all the orbits have the same dimension, Olver calls the Lie group action semi-regular.
All that being said, are there any examples of a Lie groups acting on manifolds where the action is not semi-regular? If so, what are they? Non-trivial examples would be preferred, if available.
Consider the action of $\mathbb{C}^{\times}$ on $\mathbb{C}$ by scaling. There are two orbits, the origin and the complement of the origin, and they have different dimensions. This sort of thing is pretty typical and you can come up with plenty of variations on it by considering various subgroups of $GL_n$ acting on vectors, on matrices, etc.