Stay safe & healthy
Brunei, like Malaysia, Indonesia and Singapore, has very strict laws when it comes to drugs. Drug trafficking to a certain degree has a mandatory death sentence. Other crimes, such as murder, kidnapping and unauthorised possession of firearms are also punished with death. Brunei uses caning (for males only) for rapes, as well as for less serious crimes, including illegal entry, overstaying your visa for over 90 days, robbery, corruption and vandalism. Caning is no slap on the wrist. Strokes from the thick rattan cane is excruciating and very painful. They can take weeks to heal, and even scar for life. These laws apply to foreigners as well.
During the Islamic month of Ramadhan, avoid consumption of food and beverages in public as a hefty fine in the thousands will be levied if caught doing so. Additionally, all restaurants, including non-halal ones, cease dine-in services during the day. It is best to avoid visiting Brunei during Ramadhan.
The bottom line is: Know their laws, and obey them.
In terms of personal safety, Brunei is a very safe country, on par with Japan, though you should use common sense no matter what.
If you're driving around Brunei, however, do watch out for impatient and/or dangerous drivers; take extra caution around midnight and early morning as some drivers illegally race on the roads.
Eating out is generally safe because of good food safety standards. But drink water only if it's been boiled, or bottled water. Protect yourself from mosquito bites, in this part of the world mosquitoes may carry dengue fever or malaria.