It seems that the assassination missions Lester gives to Franklin, beginning after the first heist, are the most obvious scripted occurances of stock market fluctuation. Most of these affect LCN, but some are companies on BAWSAQ.
Before each mission, invest in the target's competitor, ideally with multiple characters.
The missions are as follows:
- Hotel Assassination -- invest in BettaPharmaceuticals.
- Redwood Cigarette Assassination -- invest in Debonair Cigarettes.
- Vice Assassination -- invest in Fruit.
- Bus Assassination -- invest in Vapid after the mission. The stock will fall, but then rise again.
- Construction Assassination -- invest in Gold Coast.
Also, there is a random encounter in which you take a gentleman in a suit to the airport. If you get him there on time, he'll tell you to invest in Tickled on the BAWSAQ market. Doing so is a good idea, as this stock will climb rapidly for some time.
Additionally, as noted by other answers, performing actions against a company may increase the value of it's competitors. Examples given include destroying FlyUS planes in order to improve the stock of AirEmu and vice versa. The rivals of a company are summarised well in this answer to another question. However, I've not seen this actually work and fear it may just be a rumour at the games release that has been repeated so many times that everyone accepts it for truth.
BAWSAQ is influenced by the online community and it's likely down due to server loads. Games as popular as GTA V almost always have connectivity issues the first week or two after release so it's probably related to server load issues. Rockstar is also aware of the issues with BAWSAQ and you can check this article for updates it's current status.
Once it is is back up, if you need any help with the stock market here's a guide I found useful.
Best Answer
This does seem to be a common bug as seen in the GTA rockstar community, there does seem to be the occasional fix that buying a small number of shares that disappear means that it wont happen again, and some say that its avoidable by not using the escape key to quit the browser, however I am having difficulty finding any kind of official Rockstar support on the matter.