Let $X$ be a normal surface over a field $k$. Assume that $X$ is singular.
Does there exist a field extension $L/k$ (finite or infinite) such that $X_L$ is nonsingular?
The answer is no in general. Here's an example: $k[x,y]/(y^2-x^3)$. (To get a surface consider the example $k[x,y,z]/(y^2-x^3)$.)
But still, does there exist an $X$ such that the answer is yes?