How can one stay more than 30 days in Singapore on a tourist visa?

According to the Singapore Immigration & Checkpoints Authority (ICA) page on Extension of Short Term Visit Pass: Visitor on Short Visit for Social Purposes, last updated 30 August 2014, the requirements vary somewhat depending on whether you qualify to use the online application system, e-XTEND, or must submit an application in person.

For the online extension:

• You have not previously sought/ obtained an extension of stay on this trip
• You wish to extend your stay for not more than 89 days from the date of entry into Singapore
• Your Visit Pass has a remaining validity of at least 3 working days, excluding weekends and public holiday
• You will not return to Singapore within 5 days from the departure date of your current trip
• You will require a local sponsor if you are seeking an extension of short term Visit Pass to 89 days from the date of entry or if you are a National of [certain countries, not including India]

If you are unable to fulfill all of those requirements, you must go the counter at the Visitor Services Centre on the 4th floor of the ICA building (10 Kallang Rd, near the Lavender MRT) to submit your application, along with

• Duly completed Form 14 signed by the applicant and a local sponsor (the local sponsor must be either a Singapore Citizen (SC) or Singapore Permanent Resident (SPR) above 21 years old)
• Duly completed Form V39I signed by the local sponsor (Malaysians are not required to complete this form)
• Applicant's travel document (e.g. passport) with at least 6 months validity
• Applicant’s Disembarkation/Embarkation (D/E) card with a valid Visit Pass granted on entry into Singapore
• A confirmed onward/ return ticket (for all non-Malaysian applicants)
• The local sponsor’s Identity Card or Passport only (a copy of the NRIC or Passport's biodata page must be submitted to ICA)

Fees will apply, and in addition some applicants “may be requested to furnish a Security Deposit.” As there are further stipulations and clarifications, do read the entire page.