Singapore's historic district is home to the nation's Supreme Court, City Hall, Old Parliament House, St. Andrew's Cathedral and the Padang, which is an open field where Singaporeans often flock to play games such as rugby and cricket. You can spend 30 minutes strolling through the area or linger all morning, stopping at sights that interest you.
Once the afternoon heat hits, head to the National Museum of Singapore for history and air conditioning. It's also Singapore's oldest museum, packed with artifacts known as the museum's 11 treasures; these include the pre-colonial Singapore Stone, an ornate funeral hearse from the early 20th century and one of the first photographs of Singapore, which dates back to 1844.
Serangoon Road is a mesmerizing place to explore culture, especially in the cool of the morning. This is the site of Little India, where you'll find everything from the colorful tower of the Sri Veeramakaliamman Temple to shops selling saris, spices and Bollywood DVDs.
In the afternoon, a cab ride can whisk you to the Asian Civilisations Museum. It hosts exhibits dedicated to Singapore's history as a melting pot, with artifacts from the nation's various ethnic groups.
When it's time to wind down your day, head to the Raffles Hotel (raffles.com), a colonial establishment that has hosted the likes of Rudyard Kipling, Ernest Hemingway and Alfred Hitchcock. Here, you can shop, dine or kick back with a Singapore Sling at the Long Bar, a replica of the original.
In the morning before noontime prayers, wear long pants or a long skirt and a sleeved top to Sultan Mosque, built in 1825; you can't miss its large golden dome towering over the street. Outside, Arab Street is the heart of Singapore's Muslim district, known as Kampong Glam. Here, you can explore the shops selling local handicrafts and wares, though they're closed on Sundays.
Don't miss Thian Hock Keng Temple. It's one of the oldest Hokkien temples in Singapore, exhibiting traditional Chinese architecture -- meaning no nails and an abundance of elaborate carvings, columns and decorative pieces.
If you can brave the heat all day, Singapore has surprising pockets of greenery to explore. Start the day at the Singapore Botanic Gardens before the heat sets in. These lush gardens are home to heritage trees, sculptures and venues filled with greenery if you want to hold an event.
Later, take a whirl through Singapore Zoo, which animals such as orangutans, white tigers and Asian elephants call home. If you prefer to stay out of the afternoon heat, you can still see animals at night with Singapore's Night Safari (nightsafari.com.sg), a park dedicated to nocturnal animals. Walking trails and trams lead you to leopards, fishing cats hunting by night and Malayan flying foxes that sometimes become friendly with visitors. The park also has performances and dining to entertain you throughout the evening.