In this paper, we consider the various sets of comultiplications of a wedge of spheres and provide some methods to calculate many kinds of comultiplications with different properties. In particular, we concentrate on studying to compute the number of comultiplications, associative comultiplications, commutative comultiplications, and comultiplications which are both associative and commutative of a wedge of spheres. The more spheres that appear in a wedge, the more complicate the proofs and computations become. Our methods involve the basic Whitehead products in a wedge of spheres and the Hopf--Hilton invariants.