To travel from Singapore to Malaysia there are several options, nearly all use the two bridges that connect Singapore to Johor Bahru.
Taxis can be caught at Queen Street to go to Johor Bahru. The taxis cost $40 to go to the dropoff point in Johor Bahru CBD. The taxi will wait for 4 passengers (making the charge $10 each) or you can pay the full taxi fare ($40) to have your own taxi. To go to a specific location in Johor Bahru the cost will be $50, or more.
Buses are also an option (see my post about taking the bus from Johor Bahru to Singapore). You can catch buses at the same Queen Street station with the taxis or at the Kranji MRT (subway) stop. You can catch long distance buses in Singapore or Malaysia to further points in Malaysia.
Taking a car is another option.
Here is a webcam to show you the current status of any backups at the causeway bridge (in either direction) and Tuas bridge. The connections between the cities get very backup up on major holidays and (especially the causeway) during evening and morning rush hours and weekend traffic (leaving Singapore on Friday and Saturday morning and returning to Singapore Sunday night).
Singapore residents travel to Johor Bahru for many reasons. Two of the most popular reasons are to shop and eat at the excellent and cheap (compared to Singapore) restaurants.
You can also catch the train (which goes over the causeway bridge).
And finally you can catch a plane to further away points such as Penang, Kuala Lumpor or Kuching.
Singapore and Malaysia have announced a subway which may connect 5 subways stops (to be built) with the Singapore MRT by 2020.
Related: Johor Bahru Customs, Immigration, and Quarantine Complex (CIQ) – Singapore to JB bus services – Singapore and Iskandar Malaysia – Chez Papa French Bistro in JB CBD – Rosmarino Italian Restaurant in JB CBD
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