Are there regular ferry services between Singapore and any town or city in Borneo? If so, which towns or cities in Borneo have such ferry services to Singapore? What is the travel duration?
The Man in Seat 61 (whose website is pretty exhaustive for overland travel) says that there are no direct ferries from Singapore or mainland Malaysia to Borneo. One must instead take a ferry to Java and then to Borneo. From his page on Malaysia:
Since 1988 there have been no ferries between Singapore or mainland Malaysia and the Malaysian states of Sarawak and Sabah on the island of Borneo. The only option is a ferry from Malaysia or Singapore to Java in Indonesia (see the Indonesia page, ferry section), then a ferry from Java to the Indonesian part of Borneo, then overland the Malaysian part of Borneo.
- Ferry service from Singapore to Pulau Batam (multiple operators)
- Pelni ferry from Pulau Batam to Tanjung Priok (near Jakarta)
- Pelni ferry from Tanjung Priok to Pontianak, with a stop in Tanjung Pandan
The Man in Seat 61 says that the Pulau Batam–Tanjung Priok ferry usually runs on Wednesdays and takes 24h30m; I would expect the ferry to Pontianak to be just as long, if not longer, and you may not be able to connect to it on the same day you arrive in Tanjung Priok.
I should note that I was unable to get the Pelni website to provide me with tickets and schedules for either of their two legs. That could mean that the website just isn't working correctly, or it could mean that the ferries are not running. The Wikipedia list above is only valid as of 2016, and the majority of transportation companies drastically scaled back their operations with the advent of COVID-19. Additionally, as pointed out in the comments, Singapore is currently under fairly strict COVID-19 restrictions, so one should not expect to use this itinerary any time soon.