I find mulberry sapwood to NOT be as useless as osage sapwood. On red mulberry, I chase rings and treat it as mentioned, like lower end quality osage.
But, I have had pretty darn good success with lumpy WHITE mulberry staves, from the long branches that shoot up after people "top" their trees. THAT stuff acts, to me, like excellent white wood: top quality rock elm or something similar. Med flat, a little wide, and heat treated, it makes FANTASTIC shooters with lots of waves, dips, and bumps. Cool stuff.