It is of full rank: if the manifold is of dimension n, then the rank is n.
This is because a riemannian metric g, when evaluated at a point p of the manifold M, is a positive definite symmetric bilinear form g_p(.,.)on TpM. In particular, its matrix can be diagonalized: there is a basis e_1,...,e_n of T_pM such that g(e_i,e_j) = 0 if i and j are different. Then the rank is equal to the number of nonzero diagonal element. Suppose the ith diagonal element is 0. Then g_p(e_i,e_i) = 0, violating the positive definiteness of g_p.
More generally, a bilinear form has maximal rank iff it is nondegenerate.