isometries of the sphere

Rigidity persists a little below $C^2$, namely in the class $C^{1,\alpha}$ with $\alpha>2/3$. It breaks down for small Hölder exponents $\alpha$ in a rather strong way: any continuous embedding can be uniformly approximated by $C^{1,\alpha}$ isometric embeddings. (Key term: h-principle). This is a stronger form of Nash's $C^1$ isometric embedding theorem, which was the precursor to the h-principle. See h-Principle and Rigidity for $C^{1,\alpha}$ Isometric Embeddings by Sergio Conti, Camillo De Lellis, and László Székelyhidi Jr.

So yes, the set of isometric deformation is way too large to admit classification.