Is "今日" or "今天" more appropriate to use when speaking Singaporean Mandarin? Is it different than the one that is used in writing?
Wikipedia has a good overview on the distinctive features of Singaporean Mandarin. In Singapore's context, there is nothing unique pertaining to the usage of "今日" and "今天". 今日 is more formal and is used more often where today means "nowadays" while 今天 is usually used where today means "this present day", both in speech and writing.
The following examples demonstrate the difference in usage for these similar words:
Homosexuality is no longer viewed as a crime in today's western society.
Today's weather is great!