I am visiting Singapore in May. As a practice I always keep the passport with me wherever I travel. I think how can I show my identity if some officer ask about it. But sometimes I fell it is annoying to bring the passport. If I can leave it at hotel and travel, that make me easy. So I would like to know should someone keep the passport all the time (when travelling inside the country) even after entering in to the country? In this case Singapore.
Your passport is an important document and it should be kept safe at all times. One of the best ways in general is not to have it with you but keep it in the hotel safe.
This varies by location and in some places, it is safer with you. In Singapor, crime is very low and there should not be a problem to have it with you.
My strategy is to keep a photocopy of the important passport page with me. That copy is folded up and takes very little space. It is not an official document but has enough information to show who I am.
Note that in some countries you have the obligation to keep a piece of identification with you at all times. For a foreigner, this often means the passport but not necesarily. Any travel book of the location you are going to usually references the rule but to have the official word, consult the embassy.
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.
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
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.
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. :)