Given a copy of a stock certificate and the CUSIP # (plus any of the
other information on the certificate), how does one trace the
status/disposition of the shares represented by this certificate.

I'm not quite sure I understand you. A CUSIP is assigned to a
particular security, so all shares of a company's common stock will
have the same CUSIP.

It sounds like you're trying to find out more about a stock
certificate representing X shares of a particular stock. In that case,
you should contact the company. Assuming it's registered stock, they
should have a record by serial numbers of who owns the shares of
stock.

If it's a moldy stock certificate you found in a basement, the company
may be defunct. I'm not sure what kind of laws apply to that kind of
stale stock. Especially if the company was acquired, you may be out of
luck.