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How often do stocks become worthless?
The only thing that makes a stock worthless is when the company goes out of business. Note that bankruptcy, by itself, does not mean the company is closing. It could successfully restructure its affairs and come out of bankruptcy with a better outlook. Being a small or unprofitable business may cause a company's to trade in the "penny stock" range, but there is still some value there. Since most dying companies will pass through the penny stock phase, you may be able to track down what you're looking for by finding companies who have been (or are about to be) delisted. Delisting is not death, it's just the point at which the company's shares no longer meet the qualifications to be traded on a particular exchange. If you find old stock certificates in your grandmother's sock drawer, they may be a treasure, or they may be worthless pieces of paper if the company changed its ownership and Grandma didn't know about it.