The branching laws for the $SO(n-1)$ as a subgroup of $SO(n)$ are well known and easy to find. See for example the Wikipedia article:

https://en.wikipedia.org/wiki/Restricted_representation#CITEREFMurnaghan1938

I am having trouble finding the next most complicated example, i.e. $$ SO(n-2) \times SO(2) \subseteq SO(n). $$ Where can a description of these branching laws be found?

Edit: In particular I would like to know examples of representations of $SO(n-2) \times SO(2)$ which appear with multiplicity $1$ in any branching.