Should I always keep my passport with me while travelling?

I recommend you make a paper copy of your passport and carry that around. Leave your real passport at the hotel. I've traveled to many of the Asian countries (including Singapore) on business and never had a problem. I never needed to show it to anybody and I spent over 3 years total in Singapore, Taiwan, Japan, Korea etc.

Usually you can carry a paper copy around in your back pocket for about 4 weeks before it starts to fall apart. If you're going to be there longer than that, you might want to bring a couple copies.

Enjoy Singapore and be sure to try the Durian.

In China hotels usually "demand" to see your passport on checkin.
Maybe not in Singapore.

In China I've had to show my passport before being allowed to use public access internet connected computers. I do not know if they would have accepted a photocopy.

I always carry my passport with me. I use either

• a round the neck slung pouch slung so it sits under one arm - not noticeable even when wearing a business shirt.

• Or suspended off-centre inside my trousers in front.

• Or in a zipped front leg pocket in cargo pants when in informal travel mode. I remove the zip slider so you need to use fingernails to open it. A 500 GB HDD travels with it when flying. (If you can run from the plane only smoking then you and the photos and the passport survive. If not, you don't care.)

All these locations are "invisible" externally, comfortable for me so that I do not notice that they are there.

If you do carry it, take the utmost care with its security. I have had pickpockets attempt to "access" my bags or pockets occasionally. On one occasion I had a small bag on my back, inverted so the front pockets were against my back and a man attempted to reach in and access the zipped pocket against my back as I walked.
His confidence in himself exceeded his ability not to be noticed by me. He seemed more annoyed by the many photos I then took than I was by his actions. (Probably not a wise way to react in some cases).

FWIW I'd rate my key documents as: Passport, VISA card, baggage reclaim ticket, other including money.

Knowing where your local consulate/embassy is located would be helpful. In Singapore, both the American embassy and Australian high commission are located along Nassim Road, quite close to the Botanical Gardens. If you're from an EU country, knowing where one of the EU embassies are located would be helpful as based on my understanding, EU embassies will be there to help especially when you lose your passport.

Do ask if the hotel/hostel offers a safe deposit box, or if you could leave your passport at the reception. Carrying your photo ID like your driver's license or national identity card would suffice, especially if you intend to head to the nightclubs in Singapore. :)