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I am interested in the fares for the metro or bus system in downtown Singapore.

I search in the website but I was not able to find a proper table with fares. It seems that the prices change accordingly with the route. If so, do I need a travel card?

Can someone provide me a link or more information about this metro system.
I am interested in a weekly or daily ticket with unlimited rides. Do they exist?

The fares are based on distance. The SMRT website has pages that explain the fare structure, a table of bus and train fares, and other travel information. There's also a bus fare calculator.

They do offer the Singapore Tourist Pass, which gives unlimited transit within specified time periods. However, you'd have to visit a ticket office when you're done with it to get your deposit back.

The previous answer is great and gives all the necessary information, however I'd like to give you a more quantitative answer.

The Singapore Tourist pass costs $10/$16/$20 for 1/2/3 days and allows you unlimited travel on bus, trains (MRT) and light rail (LRT). You mention that you are interested in downtown Singapore, so most likely you can get around almost everywhere by train.

Now if you decide to just pay for tickets as you go along, you have to options: You can buy tickets each time or buy a EZ-link card. Single tickets are significantly more expensive, especially for short distances. The range of prices from the fare table is $0.77-$2.02 for card prices and $1.40-$2.50 for cash prices. The EZ-link card costs $5, so if you make short trips around town at the cheapest price, it's worth buying the card after only a couple trips. Value can be added to the card at any station and some bus interchanges, leftover value can be refunded at stations with ticket offices (for instance at the airport).

Now is it worth getting the Singapore Tourist pass? Well, assuming you're travelling around the city centre and not staying in some very far off place, your usual trip will probably hover around $1 (that will take you up to 5 km). Taking the train from the airport to the city centre is around $1.7. So for the 3-day tourist pass to be worth it, you'd have to take around 20-30 trips within those 3 days. I'd say for most people, this is really not worth it, however it might still be a good option if you don't want to think about topping up your card.

If you already roughly know where you want to travel to, you can check out the prices for those routes online. (I recommend using a website which can get you prices for both busses and trains such as this one, prices given are card fares) Note that if you change between lines or if you change from train to bus or between different busses, it'll count as a single trip.