Some stalk of non-zero sheaf , on locally ringed space, is non-zero?

Question

If $\mathcal F$ is a non-zero sheaf of Abelian groups on a locally ringed space $(X,\mathcal O_X)$, then is it true that some stalk of $\mathcal F$ is non-zero ?

If this is not true in general, then is it true if we further assume $X$ is a scheme and/or $\mathcal F$ is a quasi-coherent sheaf of $\mathcal O_X$-modules ?