I saw this question about the USA and the UK. I'm wondering what the situation is with other countries. Where can we find this type of information?
They won't know. Yet. The UK has immigration sharing treaties with the FCC countries which include the USA, UK, Canada, Australia and NZ. If you were to apply to any one of these countries you would generally be required to submit fingerprints as part of the visa process. These fingerprints could then be anonymously shared with the other FCC countries. If there was a match, other biographic data and immigration history including visa refusals would be shared.
The UK has no such agreement with Singapore. Even if the Singaporean authorities have your fingerprints, the UK does not have a treaty with Singapore which would let them access your previous refusals. Therefore, unless you tell them (or if your passport/other documents show it) there is no way for the UK authorities to tell if you were refused a visa for Singapore.
Whether or not to tell them is your decision to make. By omitting your refusal you are extremely unlikely have problems in the future. However, in the absolute worst-case scenario, you could be convicted of a criminal offence in the UK. On the other hand, by telling them about the previous fraudulent application, your credibility will be greatly reduced and you will almost certainly find it very difficult to get a visa.
As @GayotFow points out, there is no cut-and-dried answer. Goverments do share, or are able to access, data regarding immigration and visas. For example, UK and Ireland do, as do the Five Eyes alliance (Australia, Canada, New Zealand, the United Kingdom and the United States). You could enquire of the country which refused the visa, or the country to which you're applying; you may, or may not, get a response. Many governments don't like to share such information nor are many required to do so.
Again, @GayotFow: assume that everybody knows everything. Second, tell the truth in a visa application, including when asked "Have you been refused a visa for any country, including the UK, in the last 10 years?" If yes, it also asks the applicant to identify the country, type of visa applied for, and when. Failure to disclose any relevant information invites significant and, potentially, long-term consequences. Consular officials and Entry Clearance Offiers are trained to spot irregularities and inconsistencies and risking a second refusal based on deception would be a very poor decision.