  It is well known that, among closed spherical Seifert three-manifolds, only
lens spaces and prism manifolds admit several Seifert fibrations which are not
equivalent up to diffeomorphism. Moreover the former admit infinitely many
fibrations, and the latter exactly two. In this work, we analyse the
non-uniqueness phenomenon for orbifold Seifert fibrations. For any closed
spherical Seifert three-orbifold, we determine the number of its inequivalent
fibrations. When these are in a finite number (in fact, at most three) we
provide a complete list. In case of infinitely many fibrations, we describe
instead an algorithmic procedure to determine whether two closed spherical
Seifert orbifolds are diffeomorphic.
