If $X$ is a normal variety, and $p \in X$, is it true that there is a curve $C \subset X$ with $p \in C$ a smooth point (on C)?
It is obviously false if normality is dropped - take $X$ to be a singular curve. I have no specific reason beyond this to request normality, though the examples I am familiar with pass this test. I somewhat doubt that this could be true, so I am asking for a counter example.
(The vague motivation is that I am thinking about the valuative criterion for separatedness recently, and would like to understand the intuition that there are no curves $C$ with double points on a separated scheme - i.e with two centers for the same valuation on $C$. And I like DVRs, though I guess one can just take an arbitrary curve passing through the point and take its normalization to get a discrete valuation on the curves function field with some prescribed center. I am still curious about the geometric question anyway.)
The other side of this question:
Is there a (normal) variety $X$ with a singularity so "bad" that all curves passing through that point acquire that singularity?