The end of the sprouts on the potatoes I just bought are black. Should I use them?
I'd just trim them off -- as long as the interior of the potato isn't black, you're ok.
what @amanda said. Just peel off the skin too, if it's looking yucko.
On a similiar note, let's say the potatoe has green nodes(?) growing on it, as if it was sprouting, after sitting in the closet bin for a week or two...is it time to throw the taters away?
same rule - sorta - cut away anything that's green and sprouty. The rest should be fine. The green parts are full of solanin - which is poisonous in large quantities, but just unpleasant to eat in small amounts.
To add to the already good information: The green on a potato will make that area taste bitter, AND if it's a small potato, the whole potato taste bitter. I work on a small organic farm. The current trend is toward teeny tiny potatoes....When I'm sorting for market, I remove all the greenies as they will make the whole pratie taste bitter.
The "black spot" is most likely caused by a fungus. On a larger tater, cut it off. If it's smaller, toss the whole spud into the compost.