Edit: as pointed out below in comments, what I wrote originally is wrong. I will try to come back later and repair this, but if someone else has a better answer before then, I may just delete this "answer".
Since the $k$th singular value is the distance to matrices of rank $\leq k-1$, one clearly has $s_k(A+B)\leq s_k(A)+s_k(B)$. This ought to be sharp in the sense that one ought to be able to find matrices $A$ and $B$ where equality is attained for all singular values $s_1,s_2,\dots$ (and the examples can be positive and commuting, I think, but I haven't thought about this too much).
As for trying to find inequalities in the other direction, I think taking $A=-B$ kills the most naive attempts.
Edit: on rereading your question I see that this doesn't really answer it at all. Sorry!
