There are many kinds of inequalities one can obtain: In fact, the following quite general statement holds:
Let $n,m \in \mathbb{N}$, and let $q=min(n,m)$. For any norm $\| \cdot \|$ on $\mathbb{R}^q$ which is invariant under signed permutations*, and any two real $n \times m$ matrices $A,B$:$$\|s_1(A+B),\dots,s_q(A+B) \|\le \|s_1(A),\dots,s_q(A) \| + \|s_1(B),\dots,s_q(B) \|$$
This is holds since one can prove every such norm induces an orthogonally-invariant norm on the space of $n \times m$ matrices in a natural way. (see here for details).
In particular the quantity $(s_1^p+\dots s_q^p)^{1/p}$ is subadditive for any $1 \le p \le \infty$ (as mentioned also by Yemon Choi).
*such a norm is called a symmetric gauge function