What is an example of a connected symplectic manifold $(M,\omega)$, with a Hamiltonian action of $G = U(1) =S^{1}$ with infinitely many stabiliser types?
Infinitely many stabiliser types means that infinitely many sub-groups of $G$ appear as stabilisers as points in $M$.
I am aware that $M$ is necessarily non-compact.
As an simple example with infinitely many different stabilizer subgroups you may take countably many disks. Since each disk may be rotated independently with a different speed, we obtain an action of $ U(1) $ such that all the subgroups $ \mathbb Z / n \mathbb Z$ with $ n \in \mathbb N $ occur as stabilizer.
Of course the example obtained in this way is disconnected and even has infinitely many connected components. However, one can use the same strategy to obtain a connected manifold with infinitely many distinct stabilizer groups. The idea is to embed the collection of disks in a bigger space and "smear" the action between them. Details can be found in an old paper "On a problem of Montgomery" by C.T. Yang.
