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Does there exist a Riemannian metric on the $n$-sphere ($n > 2$) such that at each point some (but not every) sectional curvature is negative?

For $n=2$ it is easily seen that such a metric cannot exist.

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    $\begingroup$ If you just need one example, it is quite easy to construct (and verify) an appropriate left-invariant metric on the Lie group $\mathrm{SU}(2) \cong S^3$. $\endgroup$
    – Nate Eldredge
    Commented Jun 19, 2018 at 14:52

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Joachim Lohkamp has shown (Annals of Mathematics, 1994) that any smooth manifold of dimension greater than two admits a metric of negative Ricci curvature. So it seems that the answer to your question is yes.

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Kazhymurat's answer is definitely correct -- every sphere has such a metric.

But, much more explicitly, if your sphere is odd-dimensional, then there are even homogeneous metrics (called Berger metrics) with some planes of negative sectional curvature. Since these metrics are homogeneous, this happens at every point. It is even possible to arrange for these metrics to have arbitrarily negative scalar curvature. Besse's "Einstein manifolds" book may be a good starting point reference.

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