Let $C$ be an irreducible and non-singular algebraic curve over an algebraically closed field $k$, and let $S \subset C$ be any finite set of points. Is it true that there is always an open affine $U \subset C$, which contains all of $S$?
If $C$ is projective, this seems to be true, see this answer. This also settles the case for quasi-projective curves.
Some context: I think this is implicitly used in Serre's Algebraic Groups and Class Fields. He wants to construct another curve $C'$, such that $C \to C'$ is the normalization of $C'$, and $S$ is the preimage of the singular points. He claims that it suffices to construct $C'$ for the affine case, and glue the affines together. However I think this only works if we find an affine which contains $S$.
Every curve is quasi-projective: see Stacks 0A25, for instance. One strategy for the proof is to find an ample line bundle, take a high-enough tensor power so that it becomes very ample, and get an immersion in to $\Bbb P^r_k$. In fact, $\mathcal{O}(p)$ for $p$ a smooth point will do for our ample line bundle: $\mathcal{O}(np)$ for $n\gg 0$ will be very ample by Riemann-Roch. (I'm sure that this is recounted somewhere here on MSE or MO, but I can't find it at the moment - if you are reading this and can find the link, please put it in the comments!)