It is shown that any weakly stable Yang-Mills field of type SU(2) or SU(3) on the four-sphere must be self-dual or anti-self-dual. Any Yang-Mills field on S(n), n >/= 5, is unstable. Examples of stable fields on S(4) and S(n)/Gamma for n >/= 5 and Gamma not equal {e} are given. It is also shown that, for any Yang-Mills field R on S(4), the pointwise condition parallelR(-) parallel (2) < 3 (or parallelR(+) parallel (2) < 3) implies that R(-) = 0 (or respectively that R(+) = 0). In general, any Yang-Mills field R on S(n), n >/= 3, that satisfies the pointwise condition parallelR parallel(2) < (1/2)((2) (n)) is trivial. If n = 3 or 4, the condition parallelR parallel(2) </= (1/2)((2) (n)) implies that either R is the trivial field or it is the direct sum of a trivial field with a field of tangent spinors carrying the standard connection.