The cheapest way to get to Singapore's nearest Malaysian city, Johor Bahru, is to catch Singapore Bus Service (SBS) bus number 170 to Larkin Ter in JB. It's a two-hour ride that leaves from Queens Street bus station every 15 minutes and costs S$1.70 one-way. The bus will drop you at the Singapore checkpoint, but it will not wait, so bring all your belongings through immigration. Keep your bus ticket and hop on the next one that comes along.
Air-conditioned private buses between Singapore and Malaysia are quite comfortable. The Singapore–Johor Bahru Express runs daily every 15 minutes 6:30 am–midnight from the Queens Street bus station; the one-way cost is S$2.40. The Malacca–Singapore Express takes about 4½ hours, costs S$11 one-way, and leaves from the Lavender Street bus station seven times daily. The direct Kuala Lumpur–Singapore Express leaves from the Lavender Street station eight times daily. Several more expensive (S$30–S$47 one-way) and more comfortable express coaches travel between Singapore and Kuala Lumpur.