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
- Plenty of singers get by with unconventional voices because they can sing in tune.
  That's a damn fact. Dave Mustaine has an absolutely awful singing voice (and an awful attitude but that's a different story) but he just went out there and went wild with it and became one of the most recognizable voices in classic metal. Recognizable largely because it's awful but he manages to spin it into something people want to hear. Nobody else sounds like him and he made a living off that.
  I'd argue the same for Bon Scott and Brian Johnson of AC/DC. I'm not sure they could have made it in any other genre, but they did damn well for themselves playing good old rock and roll. It's about matching a style to your voice and then having the confidence to go out there and blaze a new trail. But I suppose that's the same for any music.
  
  I think of Brody Dalle in The Disillers. Such an harsh and unexpected voice. https://youtu.be/eFVT_n4MiKQ
- Yeah, look at tom waits.
  
  Tom Waits for no one

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!
- Thanks for the advice, and I'm happy with you improvement and confidence in your singing!

- That’s right. I think with a little training you’ll surprise yourself OP.

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.
- Thank you! I already play guitar, bass, and drums. I just want to complete the thing so I can write and sing my own songs :)
  
  Awesome :)
  Just keep in mind that you don't have to sound like anybody but you. Wanting to get technically better at singing is a worthy cause, but don't let sounding different, or even "bad", stop you from writing and singing your own songs right away.
  So many of the artists that I love (folk guys like Neil Young, punk rockers like Tim Armstrong, and so many other people across a bunch of genres) are people with pretty unconventional voices who are just confident enough to pull it off anyway.

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.
- It was pretty much a staple of my favorite genre: grunge.
  Yes!! I'm totally going for this style of voice anyway 😋 🤘

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
- I'm totally getting that!! thanks 🙂

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
- Lol, go sing in a forest.

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 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!
- Thanks for the advice and encouragement! Any tips pn how to start shopping around for a voice teacher? Do I just google it, look on social media, or else?