SINGAPORE EXPATS FORUM
Singapore Expat Forum and Message Board for Expats in Singapore & Expatriates Relocating to Singapore

Little India is just next to Boon Keng.

Wide selection

or you could try

Lock & Store http://lockandstore.com.sg/ if you are in the Holland or Orchard area, or

Safe2Store http://www.safe2store.com.sg/ if you are in the Western part of the island.

Try calling some of the local self-storage operators. They don't provide moving services themselves but because their customers do, they may be able to recommend you a "good, cheap and reliable" mover.

Some common rooms in newer condos do have. But if you are renting one in a HDB apartment, almost all won't have it attached.