A halal screen tells you whether a company's shares are permissible to own. It does not tell you whether every way of trading them is permissible. Three practices come up most often.
Margin trading
Margin means borrowing from your broker to buy more shares than your cash allows. The broker charges interest on the loan, and interest is prohibited. Most scholars therefore do not accept margin trading, even when the shares bought are halal.
Short selling
In a short sale you sell shares you borrowed, hoping to buy them back cheaper. Most contemporary scholars, and the International Islamic Fiqh Academy, do not accept it, because you sell something you do not own and the arrangement usually involves a loan at interest.
Options and futures
These are contracts about the future price of an asset. Many scholars and fiqh councils have ruled against conventional options and futures because of uncertainty (gharar) and speculation. Some alternatives have been proposed, and views are not identical, so ask a scholar before using them.
What this means in practice
Buy fully with your own cash, hold the shares you own, and avoid borrowing and speculative contracts. Check the stock itself and also how you intend to trade it.
This is a summary of common views, not a fatwa. Brokers' offerings differ by country, and Islamic accounts without interest are available from some brokers.