  We study degenerate elliptic operators of Grushin type on the $d$-dimensional
sphere, which are singular on a $k$-dimensional sphere for some $k < d$. For
these operators we prove a spectral multiplier theorem of Mihlin-H\"ormander
type, which is optimal whenever $2k \leq d$, and a corresponding Bochner-Riesz
summability result. The proof hinges on suitable weighted spectral cluster
bounds, which in turn depend on precise estimates for ultraspherical
polynomials.
