This note is on spherical classes in H * (QS 0 ; k) when k = Z, Z/p with a special focus on the case of p = 2 related to Curtis conjecture. We apply Freudenthal theorem to prove a vanishing result for the Hurewicz image of elements in π s * that factor through certain finite spectra. Either in p-local or p-complete settings, this immediately implies that elements of well known infinite families in p π s * , such as Mahowaldean families, map trivially under the unstable Hurewicz homomorphism p π s * ≃ p π * QS 0 → H * (QS 0 ; Z/p). We also observe that the image of the integral unstable Hurewicz homomorphism π s * ≃ π * QS 0 → H * (QS 0 ; Z) when restricted to the submodule of decomposable elements, is given by Z{h(η 2 ), h(ν 2 ), h(σ 2 )}. We apply this latter to completely determine spherical classes in H * (Ω d S n+d ; Z/2) for certain values of n > 0 and d > 0; this verifies a Eccles' conjecture on spherical classes in H * QS n , n > 0, on finite loop spaces associated to spheres.