Of course, this is by no means that Muslims are not allowed to own non-Syariah-approved shares. They are just discouraged to do so. As a non-Muslim, I do not refer to the list when I invest in the stock market.
(By the way, “share” is the British English term. The American equivalent is “stock”.)
I am not clear of how the Islamic institutions select the Syariah-approved shares. I do know that conventional banking in which lenders earn interest is considered haram (illegal). It is, therefore, not surprising that virtually all banking shares are not Syariah-approved.
Two airlines companies are listed in the stock exchange – Malaysian Airlines (MAS) and AirAsia. The latter is Syariah-approved while the former is not. MAS serves alcoholic drinks to the passengers, and alcohol is prohibited among Muslims. Perhaps this is the factor which contributes to its failure to earn approval. AirAsia is no-frills airlines.
But seriously, I believe that AirAsia does not deserve the status. The reason : its female cabin crews wear short skirts!