How would someone with a terrible singing voice go about learning to sing?

Singing coach would be the best answer. But outside of that I'd say the first step would be to train your ear. Pitch is more important than the quality of your voice. Plenty of singers get by with unconventional voices because they can sing in tune. And even someone with the most beautiful sounding voice will sound terrible if their pitch is bad. Developing a better sense of pitch is possible, even on your own over time. Look up ear training voice exercises.

Edit: spelling

I lived in a town with a decent music school at the university. I put up an ad on craigslist and just laid out what I wanted: a coach to help me get better control of my voice. I briefly explained my expectations: get better so I can reclaim my love of singing, pure personal enjoyment.

Since I was realistic and reasonable in my expectations, I quickly found a senior student with ambitions to open a school after graduation. We met weekly for a couple of months and the rate was reasonable.

The key lesson I learned is that my natural range is a bit higher than I thought. I got better and I don’t particularly worry anymore what people think of my voice. It feels good to sing knowing you are approximately in tune. It’s mostly bawdy drinking songs anyway!

Get a voice coach. They can teach you techniques to improve your range and tone.

As the spouse of a singing teacher... Take lessons from a trained professional (someone that has a degree in vocal pedagogy and / or vocal performance) and practice regularly. Don't expect to be amazing overnight. It may take years of lessons and daily practicing to just get to a level where people don't cringe when you try to sing something.

The first thing to try is matching pitches: play a note on some kind of instrument or app, listen to it and try to sing that same note. Do it again and again and again, multiple times every day until you feel that you have no problem doing it. This is the starting level of "ear training" and its a hugely important part of singing and playing instruments. Music is just as much about your ears as it is your instrument.

It's possible that some people are just naturally bad at matching pitches and will always be a little bit out of tune, but if that's the case for you even after a few months of ear training don't let that stop you from taking part in music, because there are plenty of iconic artists in the pop/rock/folk/blues/hiphop world who technically can't sing for shit and people still love them. In other words, practice and train your voice as much as you can, but also learn to embrace your unique voice and make it work for you.

Every choir teacher I've had, every musician I've seen talk about this subject, and I do myself believe anyone has the capacity to sing well. It is a skill, and it can be developed.

Seek vocal lessons. You can even find some on YouTube for various styles. How to croon, how to demon yell, even throat singing!

And to be perfectly honest: Some of my favorite singers are those who have been wildly criticized for their "bad" voice. It was pretty much a staple of my favorite genre: grunge.

One option you could try is the recently-released singing feature in the Ultimate Guitar app:

https://help.ultimate-guitar.com/en/articles/6873192-what-is-ultimate-guitar-sing

Singing with someone makes you learn how to adjust your voice to the tune on amateur level. Try camping or any other context you can sing with somebody. Also listen to different kinds of vocal and try to imitate them.

i dont know about the possibility, but i would first look out for songs that you like and songs where the singer have a close voice (i am a failure in that point)
And if you cant risk dealing with neighbours download the music on the phone, print the text and sing in a forest or something
Another thing is to make the song louder than your voice at the start, that way the song can guide you until you are save enough to lower the volume

And stay hydrated

I think this are more general tips and i cant guarantee that it helps

I was in the same situation as you. I played guitar, bass, and drums and wanted to sing because I wanted to start a band and couldn't find a singer. I joined my high school choir for free lessons and dedicated 5 day per week practice sessions. It helped a lot. I have now been singing for about 15 years, and I still have a lot of room to improve, but I can carry a basic tune pretty well. I'm nowhere near good enough to be a professional singer, but I can express the ideas in my head just fine.

Some good advice here. I’d add that you should figure out what your range is, don’t torture yourself trying to hit notes outside of it.

There is absolutely no substitute for a live, in-person voice teacher to guide your practice. Shop around—take a first lesson with a few options and try to evaluate who's got the best technique and pedagogy. Also, this is a great goal and very achievable—go for it!