Could anyone familiar with Singapore please tell me what is the name of a common textbook used for Math in Singapore's secondary schools? I am looking for topics covered for ages 14-18. I am very interested in what I have heard about Singapore's math education, so it's mostly to see their approach since I also do not like the vast majority of American math textbooks.
I'm only moderately familiar with Singapore, but here is what I know.
There was an older textbook series for ages 12 to 16 called "New Elementary Mathematics: Syllabus D" Books 1 to 4. I doubt all schools used it, but I think it was from the same publishing company that put out the elementary-level books that were eventually sold in the U.S. I'm not sure what they use now.
The series was formerly sold in the United States. Here is a review by, I assume, a homeschooling U.S. parent: https://cathyduffyreviews.com/homeschool-reviews-core-curricula/math/math-grades-7-8/new-elementary-mathematics
If you do a search for "Ordinary Level Mathematics" (ages 12-16), "Additional Mathematics" (ages 14-16) and "Advanced Level Mathematics" (ages 16-18), you'll turn up some materials used now.
See here for the official curricula at O-Level: https://en.wikipedia.org/wiki/Singapore-Cambridge_GCE_Ordinary_Level
For A-Level, you ought to be able to find syllabi by searching for H1 Mathematics, H2 Mathematics, H2 Further Mathematics and H3 Mathematics. According to this thread, textbooks aren't widely used at this level, but there are textbooks for similar exams offered by "Cambridge International" up to the "Further Maths" level: https://www.reddit.com/r/SGExams/comments/jzz91o/a_levels_h2_maths_and_further_maths_textbook/
The entire Singapore curriculum at the secondary level is modelled on older curricula in use in England (probably about pre-1985). So if you want more textbooks like that, you can look at the books in use in England in that period. They were very different from the ones used now because at that time the top 25% had their own curricula and were sometimes educated in separate schools. Beware of pre-1970 books, though, as they use Imperial units and non-decimal (!) currency.
If you want to see what was typical in the early 1980s for ages 11 to 15 or 16 in England (in the top 25%), there are some books on the Internet Archive called "New General Mathematics for West Africa" by Channon and McLeish Smith. These are basically the same as the British textbooks but with "London" replaced with "Lagos" and pounds with local currencies, etc. (I'm not saying these are the best textbooks, but at least you can look at them.) For "Additional Mathematics" for advanced students at ages 15-16, see "Pure Mathematics: A First Course" by Backhouse, also on the Internet Archive. This was also a first-year book at Advanced Level (age 16-17) for those who hadn't taken Additional Mathematics. I believe all of this is quite similar in approach to the Singapore curriculum.
The main difference between the Singapore syllabus and most English syllabi about pre-1980 appears to be that some pure geometry was taken out and some vectors and transformations added in.
Edit: Here are some links to official syllabi and additional information provided to me in the comments.
2023 syllabus for H2 Mathematics (Pure Mathematics : Probability & Statistics = 7:3)
with Additional Mathematics as prerequisite
2023 syllabus for H2 Further Mathematics (Pure Mathematics : Probability & Statistics = 3:1)
with H2 Mathematics as corequisite
2023 syllabus for H3 Mathematics (Pure Mathematics)
with H2 Mathematics as corequisite
2023 syllabus for H1 Mathematics (Pure Mathematics : Probability & Statistics = 4:6)
restricted to non-H2 candidates; a proper subset of Additional Mathematics and H2 Mathematics
The O-level (Elementary) Mathematics course takes place over four years (13-16yo), and the Additional Mathematics alongside it over the final 2 years (15-16yo); they are designed to occur in parallel, so the texts don't overlap, although some pre-requisite stuff from EM does get reviewed at the beginning of the relevant AM chapters, and complementary topics across the two subjects occur alongside each other.
About half of all 16-year-olds take Additional Mathematics, a minority which will go on (17-18yo) to take H1 Mathematics instead of H2 Mathematics, which will have been taken by half of all the college students (most of the STEM ones and some of the non-STEM ones). H2 Further Mathematics is a very unpopular course; almost nobody takes H3 Mathematics.
About 30% of H2 Mathematics overlaps (at different depths) with Additional Mathematics, and about 60% of the latter overlaps with the former.
There are no common textbook(s) widely used across schools.
Instead, each individual school/teacher will usually have their own notes (that they guard jealously but are usually leaked out).
Check out this "Holy Grail" collection of resources (mostly exam papers but also has some notes).
Anonymous' answer is excellent and comprehensive. To add on to that, as a Singaporean, I can share my experience with the education system here.
For ages 13-16 (secondary school) different teachers tend to have their own teaching materials, whereas for ages 17-18 (junior college) notes tend to be issued on a per-school basis, referred to as "lecture notes". To my knowledge, there is no standard textbook widely used in the country for any secondary school / junior college level math education.
However, it should also be noted that Singapore's education system has different "tracks" for students of varying academic levels / interests, most notably for ages 13-16 there are the Normal (Technical), Normal (Academic), and Express tracks, whereas for ages 17-18 there are Junior Colleges and then there are Polytechnics. Details may vary widely based on the different academic tracks, or based on the school, or even based on the individual teacher. It may be the case that a particular teacher chooses to work with a certain textbook for example, though most in my experience do not.
To answer your underlying question of what is taught in Singapore's math education system, I suggest you to Google "Singapore O Level math syllabus" and "Singapore A Level math syllabus". (Edit: I realised these were already mentioned in Anonymous' answer.) O Levels are a national exam generally taken at the end of the year for 16 year olds, and the A levels are generally taken for 18 year olds. The syllabus documents are made publicly available by the national exam board, the SEAB, online. All teachers in Singapore teach based on these exam syllabi and many have an exam-centric approach, so this is a very good guide to what is actually taught in classrooms.
If you have further questions, feel free to ask in the comments.