First, the very fact that things change with regularity is interesting, and therefore worth investigating. In the case of riemannian geometry this is quite well understood, since $C^2$ is what is needed to introduce curvature. Yet, one can isometrically embed a flat torus in $\mathbb{R^3}$ by a $C^1$ map, and this seems worth understanding (and definitely not "garbage").

Second, there are some cases where the "generic" regularity is weak: if you consider the stable and unstable foliation of the geodesic flow on a negatively curved compact surface, then it is in general only $C^{2-\varepsilon}$. A very nice theorem (by E. Ghys, if I remember well) says that if these foliations are $C^2$, then the surface has constant curvature.

First, the very fact that things change with regularity is interesting, and therefore worth investigating. In the case of riemannian geometry this is quite well understood, since $C^2$ is what is needed to introduce curvature. Yet, one can isometrically embed a flat torus in $\mathbb{R}^3$ by a $C^1$ map, and this seems worth understanding (and definitely not "garbage").

Second, there are some cases where the "generic" regularity is weak: if you consider the stable and unstable foliation of the geodesic flow on a negatively curved compact surface, then it is in general only $C^{2-\varepsilon}$. A very nice theorem (by E. Ghys, if I remember well) says that if these foliations are $C^2$, then the surface has constant curvature.