Let $M_n$ be the space of real $n \times n$ matrices, and let $|\cdot|_p$ be the $p$-Schatten norm on $M_n$, for $p \neq 2$. For $A \in M_n$, $|A|_p=\big(\sum \sigma_i^p(A)\big)^{1/p}$, where $\sigma_i(A)$ are the singular values of $A$.
Is $| \cdot |_p$ differentiable at any non-zero point $A \in M_n$? Is it smooth there? If not, at which points is it differentiable?
I think that $| \cdot |_p$ is $C^{\infty}$ when restricted to $GL_n(\mathbb R)$, since $|A|_p^p=\text{tr}(|A|^p)$: The map $A \to |A|$ is smooth on $GL_n(\mathbb R)$, I guess that the map $A \to A^p$ is smooth on the domain of symmetric matrices. (Is it really? I am not sure).
Let $B$ be a $n\times n$ (real) symmetric matrix. Then $B \mapsto \lambda_i(B), \ \forall i$ is smooth on the region where $B$ has simple spectrum (no repeated eigenvalues). See "4. Eigenvalue deformation" in https://terrytao.wordpress.com/2010/01/12/254a-notes-3a-eigenvalues-and-sums-of-hermitian-matrices/
Now, we have $\|A\|_p = (\mathrm{Tr}((A^\mathsf{T}A)^{p/2}))^{1/p}$. So, $\|A\|_p$ is smooth on the region where $A^\mathsf{T}A$ has simple spectrum (no repeated eigenvalues).