In developing countries in SE Asia, next-day laundry service for a few dollars per kg is common. Are there any in Singapore, even if more (but not too) expensive? I would prefer that over coin-operated laundry. Hotel Indigo Katong doesn't seem to have free laundry.

You could try these guys: https://piing.com.sg/tourist
I've used them... generally ok but sometimes bad weather/heavy rain can result in pick up/drop off delays.
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These guys are $7 Kg, don't know what the turn around time is.

https://lagoonatkatong.wixsite.com/katong-v

There are couple of threads about this subject here, search them.

In general - keep your expectations low. If your laundry is mostly shirts and underwear, you might find out that it is cheaper to buy new than send them for wash.

P.S. Semi-philosophical remark: if you travel with clothing that can be washed in a hotel sink with cold water and dried overnight at bathroom or with hairdryer/iron, this problem goes away.

True. However, I've had some favorite t-shirts elongate due to presumably insufficient hand-drying before hanging (even though I tried). My days of wearing exclusively wick clothes during travel is behind me (though it was convenient).

Another problem I've had specifically in Singapore is that if clothes aren't hang-dried immediately (after washer spinning and maybe hand washing), they often get a foul smell reminiscent of someone who haven't showered in a week.

Yup and the weather in Singapore can be fickle recently. You could wash your clothes now and it be blazing hot. Half an hour later it could be raining for hours.
I've also had this problem in other tropical climates.

I've actually found one of the most effective ways of drying clothes to be in air con, usually I do it over night in the bathroom and they dry fairly well. Obviously this is less effective for heavy clothes.

You could try Joo Chiat City - Dry Cleaning and Laundry Services. It's close by the hotel but not 100% certain if it's does next day.
There's also for the love of laundry, which used to have a next day service, but on the website it's suspended (not sure if that's still the case or if it's just not been updated). I have no affiliation but some ex colleagues had used it before and said it worked ok for them.