Oop, can't believe I missed this thread.
For the record, there are TWO causeways linking Singapore to Johor state, the southernmost state in Peninsula Malaysia:
In
RED is the original Singapore-Malaysia causeway (aka "The Causeway", "Woodlands", "Iskandar") connecting Woodlands district in the North of Singapore to the city of Johor Bahru/Baru, capital of Johor/Johore state (blame the British for the nonstandardized Romanization of the Malay place names).
It is older and (usually) more crowded than the other option...
In
ORANGE is the SECOND Singapore-Malaysia causeway (aka "The 2nd link", "Tuas", "Abu Bakar") connecting Tuas district in the West of Singapore to the city of Nusajaya, a very (<10 years) new development and developmental region in Johor.
It is newer and (usually) less crowded than the old causeway.
In
GREEN is Johor Baru city aka "JB". It is a lawless border town almost exactly analogous to Tijuana, with the expected massage parlours, illegal gambling, overcrowded streets, dilapidation and urban decay, and violent crime, but much less weed. Generally not a pleasant place, it does have some very expensive and extremely nice parts rivalling other rich areas in the region.
In
BLUE is Nusajaya, the Sultan's very clever (IMO) attempt to parlay some of his billions of personal wealth into more billions of personal wealth by draining swamps and cutting down palm oil plantations to throw up shopping malls, condos and amusement parks to handle spillovers from overcrowding and refugees of pricetag-shock from Singapore. The area is significantly overbuilt, and there are plenty of bargains to be had housing-wise. Basically a Malaysian Hoboken to the Singaporean Manhattan.
WRT to OP's original question, the easiest and most hassle-free option for travel between the countries is the
Causeway Link bus from Jurong and Boon Lay in Singapore to Gelang Patah and Bukit Indah in Malaysia, via the
BLUE Tuas link from 6 am to 10 pm daily. It is SGD$4 SG->MY and 4 MYR MY->SG, takes 45 min in light traffic, up to 4 hours in jams. There are of course many other buses, coaches, taxis, trains and private car services which WikiTravel is a good resource for, but I find the CL bus cheap, fast and hassle-free.
Jams occur during all commuter rush hours, 6am-9am heading into Singapore, and 4pm-8pm on weekdays heading into Malaysia. On weekends there is a perpetual jam into Malaysia. Most of the jams are due to Singapore, which not so subtly attempts to discourage Singaporeans from living in Malaysia by only running skeleton crews at the immigration checkpoints. Typical traffic scenario as I am writing this post Friday AM:
It will be the inverse this evening as people leave work in Singapore and go to Malaysia.
Make your travel outside these times.
If OP is just staying a month and not needing to work, I would strongly recommend STAYING IN MALAYSIA. (I realise this is already past his June date, but hopefully this post is useful for people in the future).
I personally have to spend ~4 days a week in Singapore. I leave Nusajaya Saturday afternoon, come into SG, then leave SG Wed or Thu at 2 pm exactly. If you have to work in Singapore daily for a contract or something, it might be better to stay there in terms of psychic damage.
Disclaimers: I HATE Singapore, as only a native son and world traveller can. I believe that all of Fathom's excellent points about the Chinese in China he wrote here:
thread-3678-...#pid230685
Apply just as much to the Chinese in Singapore, with the added "bonus" that they imagine themselves able to speak English, which adds another layer of completely unjustified self-aggrandizement.
I really enjoy Malaysia, especially since as a foreigner I am outside their utterly corrupt and morally outrageous political system.
I have some properties in Nusajaya I am trying to move.
Always happy to talk to RvFers with any questions they might have about this part of South-East Asia for free.
Don't fall for scams! Both MY and Singapore are awful for this and view white people as targets ripe for plucking.