Local Buses to JB, Malaysia - There are many buses plying the route Malaysia-Singapore daily through Johor. Singapore Bus Service (SBS) buses and the Malaysian Causeway Link Yellow buses is available almost every 15 minutes. These buses takes you across the causeway to Johor Bahru (JB) city centre or to the Larkin Bus Terminal, Johor.
Some express buses companies KL-Singapore:

•   Aeroline --- executive coaches KL to Singapore and Penang to Singapore
•  First Coach (Novena MRT) -- Singapore to KL; from KL (Bangsar/Bandar Utama) to Singapore Novena - above MRTstn or the Plaza Singapore
• TranStar ---- executive coaches from Singapore to KL and other major towns malaysia
• Grassland Express Singapore  --- Singapore to Alor Setar, SP
• Airebus - Singapore-KL  
• konsortium.com -- from major cities Malaysia to Singapore 
• Transnasional -- from major cities Malaysia to- Singapore 
• Plusliner & nice -- from major cities Malaysia to Singapore

(note: NOT advisable to travel this during weekdays or non holidays because of lack of passengers.
......Check links to bus companies and details on traveling by bus or coach to Malaysia at Malaysia to Singapore pages on myMalaysiabooks

WHERE TO STAY
Hotels in Malaysia - check and compare rates
Hotels in Singapore - compare rates from multiple websites