If $X\rightarrow S$ is a morphism of schemes, and $X\times_{S}C\rightarrow C$ is flat for all morphisms $C\rightarrow S$ from a (smooth) 1-dimensional scheme, is $X\rightarrow S$ flat?
(Is there any criterion for flatness that only involves restrictions to 1-dimensional subschemes?)
The valuative criterion for flatness (in the reduced Noetherian context) is in EGA IV. A statement and reference can be found on p.52 of Neron Models (a scan of which is available here).