The mandatory death penalty exists in Singapore for certain capital offences, including murder and drug trafficking. There are severe penalties for all drug offences in Singapore. Trafficking is defined by possession of drugs above a certain amount (e.g. 500g in the case of cannabis).
Penalties for visa overstayers include fines, imprisonment and corporal punishment (the rattan cane) depending on the length of overstay. A wide range of lesser offences such as "outrage of modesty" (inappropriate behaviour by men towards women) carries corporal punishment (the rattan cane).
Driving under the influence of alcohol is a serious offence in Singapore, and the traffic police regularly carry out breath tests. Sentences for driving whilst under the influence of alcohol can include a fine or imprisonment. The Singaporean authorities will prosecute cases of air rage within their jurisdiction.
On-the-spot fines are common. You will be fined for smoking in any public place or indoor restaurant, for chewing gum on the Mass Rapid Transit (MTR) system or littering.
You should be aware that thorough checks are being conducted on departing travellers' vehicles at checkpoints in Singapore. This may include fingerprint scanning.