Nut Shell on the compost heap?

You could since they're plant material, but they'll be a long while composting. I'd suggest making a small compost spot just for nut shells if you've got a lot of them. No point just digging them in either because I'm still digging up hidden nuts the squirrels planted years ago and forgot. Only benefit was that the acorns sprouted and I'll be moving them to my fence line to keep growing.