Singapore is about 140 km north of the equator, and it does not extend far enough south to reach the equator.
Going south of Singapore, one reaches Indonesia, and one has a lot of choice there.
Pontianak, on Borneo, has 570 thousand people, a big seaport, and is at the equator.
Padang, on Sumatra, has 1 million people, a big seaport, and is 110 km south of the equator. Going north, one gets to Bukittinggi, with 120 thousand people, and only 33 km south of the equator.
Balikpapan, on Borneo, has 740 thousand people, a big seaport, and is 140 km south of the equator. Going north, one gets to Samarinda, with 840 thousand people, and only 46 km south of the equator.
There are some other cities that I could consider, like Palu and Gorontalo, but they have smaller seaports.
These nations all have land area at the equator: Indonesia, Somalia, Kenya, Uganda, DR Congo, R Congo, Gabon, STP, Brazil, Colombia, and Ecuador.
Of these, I think that these are the best hosts for a space elevator: Indonesia, Kenya, Uganda, Gabon, Brazil, and Ecuador.