I have seen on the French Wikipedia that Ehresmann's fibration theorem is stated with the assumption that everything is $C^2$. (On the English Wikipedia, the assumption is smooth, which I suppose means $C^\infty$)

[Théorème de Ehresmann](http://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Ehresmann)

[Ehresmann's Lemma](http://en.wikipedia.org/wiki/Ehresmann%27s_lemma)

1. Does anybody know of a counterexample in the case where the smoothness is only $C^1$? 

2. I am specially interested in the case where the domain of the submersion has dimension 2 and the range dimension 1. I suspect that the theorem holds in this case (select one fiber and build a local fibration-trivialization around it by patching the x-coordinate of local submersion-trivializations where level curves would be horizontals in $\mathbb{R}^2$). Is that already proved or disproved somewhere?