Don't use your left hand for greeting, gesturing, giving something to, or eating with a Malay, Indonesian, or Indian person -- it's the hand traditionally used for toiletry purposes. Refrain from kissing or touching the opposite sex, as some communities might be offended. Note that it's common for men, particularly from the Indian subcontinent, to hold hands or interact affectionately.
If you're invited for dinner by Chinese friends or business acquaintances, leave some food on your plate to indicate that your host's generosity is so great, you can't eat another bite. At a formal meal, rice will often be served amongst the final dishes rather than at the same time as the meat and vegetable dishes. Don't pile your plate full of rice at the end of the meal. Your hosts may think that they didn't provide you with enough food. There's no shame in asking for a knife and fork instead of chopsticks. Hindus are often vegetarian, Muslims don't eat pork and must abide by strict food preparation guidelines, and some Chinese may be devout Buddhist vegetarians. Check on dietary preferences before dining with multicultural company in Singapore.
It's mandatory that you remove your shoes in places of worship. You should cover up your arms and legs in Indian temples and Muslim mosques. For women, a head covering is advisable. Use extreme caution when visiting mosques, perhaps seeking the permission of locals or the nearest person in authority to enter, and then ask where you may walk and what you may do. There may be areas where you aren't permitted to go, particularly if you're a woman.
Don't litter: it's against the law and you can be fined S$1,000. Chewing gum stuck in train doors was blamed for the shutdown of the subway system more than ten years ago, and as a result the government banned it from being sold in Singapore. You may bring it in for personal use, but be sure to dispose of it properly.
Smoking isn't permitted in public service vehicles or in most buildings. You can be fined up to S$1,000 for smoking in prohibited areas. Most restaurants and some bars have outdoor terrace areas where you can light up.
Don't publicly criticize Singapore, its politics, or its leaders, and refrain from jokes about them unless you're sure of your company -- Singaporeans can joke all they want, but you're an outsider and a guest. Share your opinions only when asked and, if negative, in a gentle manner.
Bring along a stash of business cards -- everyone exchanges them, even vacationers. If you don't have business cards, consider having personal cards with your home contact details produced instead. Offer your business card using both hands with the card facing the recipient. Likewise, when a card is offered to you, accept it with both hands and make it a point to read the card. This shows your respect for the person's title and position. If you're dealing mainly with Chinese, it's not essential but is much appreciated if you make the effort to have a Chinese translation on the reverse side of your card.
Should business contacts visit you, whether in your hotel or the office, the first thing to do is offer them something to drink, either water, tea, or coffee; you can expect the same courtesy automatically when you visit their office.
You should never separate a Singaporean from his makan (food). Avoid scheduling meetings between noon and 2 PM; if it's lunch time, make sure there's lunch, or break to go out for lunch. Be mindful that Friday is the holy day of Islam, so practicing Muslims might require a longer break to fulfill their religious obligations. Similarly, during the fasting month of Ramadan, Muslims aren't supposed to eat anything from sunrise to sunset. Scheduling long meetings in the afternoon during this time may be counterproductive.
Meetings can be very long and often inconclusive because the final decision makers are often considered too senior to attend, so be prepared for multiple meetings to achieve simple goals. Find out who's attending from their party and be sure to include people that are of equal standing in your group. Introduce the most high-ranking people to each other first and seat people of the same rank opposite one another. Listen carefully and be subtle in your approach. Confrontation is a no-no and Singaporeans couch discussions so that no one loses face. As a result, you'll hear polite vagaries rather than an outright "no." Group dynamics prevent any member from speaking frankly about the performance or actions of their colleagues, let alone their boss. Using language such as, "Yes, I can see your point, but I was thinking that as a group we could try another way." is highly effective. Expressing anger or frustration is considered a sign that you don't have the maturity to be trusted. Keep in mind that politeness is an important business tool.
You should accept invitations to social events, which invariably involve food. A successful business relationship may hinge on your social relationship and whether you've shown a willingness to try to understand Singapore better. Typically, spouses are neither present nor invited if business is to be discussed. The unwritten rule is that if you invite someone (and his party) to lunch or dinner, you're paying the entire bill. Singaporeans ordinarily don't split bills, rather they'll trade favors and take turns paying. You may witness a theatrical tussle between guest and host as they fight to pay the bill -- this is an expected ritual -- but in the end, one or the other pays the whole lot. It's an important matter of preserving face. Also, showy late-night entertaining is customary among Chinese businessmen, so prepare for sleep deprivation. You may have to endure not only full 10-course show-off dinners, but also rowdy nightclubs (possibly with clinging and wallet-sapping "hostesses"), where premium brandy is tippled like water, if you really want to get that contract.
For a more exhaustive cultural briefing, pick up JoAnn Meriwether Craig's Culture Shock! Singapore and make it your "bible."