Praise be to Allah.
There is nothing wrong with selling or buying shares if they are shares in permissible companies. If it is an agricultural company, for example, which produces permissible agricultural products, then it is permissible to buy and sell shares in it; the same applies to companies which deal in real estate, manufacturing, etc.
It is permissible to buy the shares of other people and pay them the price immediately, so that one will not be selling a loan for a loan.
Excluded from this are companies which deal in haraam things, such as selling alcohol, tobacco and music tapes. It is not permitted to have shares in these companies, or to buy these shares. If the company openly deals in ribaa (usury, interest), then it is not permissible to deal with it. but if a manufacturing, agricultural or business company is compelled to deposit its money with the bank in order to protect it from being lost or stolen, then it is permissible because of that necessity for the owners of the company to get rid of the interest which the bank gives them on their deposits. Companies should also avoid using riba-based loans in order to set up production lines and factories; that should be achieved by using cash and the money of the shareholders.
And Allaah is the source of strength.