The faucet aerator broke off with the threads still in there. I can't dig it out to replace it with a new one. Any suggestions?

Depending on the style of fuacet you have, you may be able to replace just the spout. If not, I'm afraid you will be better off replacing the complete faucet. Hope you can get better suggestions than mine. I look forward to any easy way to remedy this problem. Good luck, and please let me know how things work out.