If its worth .001, it is not worth 0. However, it likely should be.
When a company goes belly up and the stock is effectively 0, sure you can hold it. But as long as you own it, you can't write it off. So yes you can hold it until it is truely 0.
I had one of those once.... a broker did a courtesy buy for $1 for all that I owned. This allowed me to write it off. To write it off, you need to sell it or be able to prove that it is worthless to the IRS.
jb