Before online brokers and personally-directed accounts, holding a physical stock certificate was a necessity, as this was the only way to authenticate stock ownership. This is not the case anymore. Although you may not need to hold a stock certificate, you may request one. The corporation you are holding stock in issues stock certificates, and you can get your certificate either directly from the issuing corporation, or by contacting your broker who may get the stock certificate on your behalf.
Detailed on the stock certificate itself will be your name, the company's name and the number of shares you own. There also will be a seal of authenticity, a signature from someone with assigning authority authenticating the certificate and either a CUSIP or CINS number. Currently, stock certificates are seen more as collectibles and souvenirs than actual records of ownership.
On the other hand, corporations may not have an interest in sending all shareholders stock certificates, although they are required to by law if requested. Issuing stock certificates increases clerical work. Moreover, corporations must record all persons who hold their stock at the end of the trading day in their computer systems. Although this information can be received from all brokerage houses, this is still something the corporation must keep track of.
Traders and short-term investors may find the issuing of stock certificates burdensome, as these stock holders may buy and sell this corporation's stock over a few days.
To learn more about what to do with old stock certificates, read Old Stock Certificates: Lost Treasure Or Wallpaper?