When I was in my 20s, I was a genuine jazz geek musician. A "muso" who wanted to be one of "the cats." Would you ever see Ron Carter or Eddie Gomez singing? No, because they completely earned their positions by playing bass. Asking them to sing would be an insult. That's the crowd I saw myself belonging to (in orientation, not talent level).
Then I started to notice simple, solid country and folk bassists getting lots of good gigs because they were real singers.
For a lot of my playing years, I've worked with *really* good singers ("recording artists") who would call on me to sing harmony, or sometimes just say, "Ok, I want everyone to join in on this!" These were clearly people who had been singing since they were joyously running around the house as toddlers, using a hairbrush as a microphone. They weren't dour, introverted perfectionists like me.
A well-known singer-songwriter once told the leader of a band I was in, "Don't make your bass player sing [background vocals]."
But finally, I decided to stop running away from it and meet the challenge. The first time I sang lead on a couple of blues tunes at an open mic, I was scared to death. I was terrified. But it does get easier the more you do it. I wouldn't say that I'm a good lead singer, but I do stay on pitch, and I'm slowly getting better at "selling" songs and actually injecting some personality and emotion. In my current covers/originals band, I usually sing lead on only three or four songs per night.
My tips for brave beginners:
1) For each song you want to attempt, find a good key for your range.
2) Find songs which have bass parts which can easily work on autopilot, because you *are* going to be using 90% of your consciousness on singing. It's actually kind of exhilarating to be totally caught up in the singing, hear "someone" backing you on bass, and realize that the someone is you! Let go of the "but... as the bassist, I'm the captain of this ship" self image. You're going to have to trust the others not to let everything fall apart.
3) Stand up straight and keep your lungs filled with air.
4) I like to use earplugs to protect my hearing, but they also help you hear the notes you are singing (like putting your fingers in your ears). This *really* got me past the worry of getting close to the mic, opening my mouth, and having some awful, off-key stuff come out. You can hum your first notes off-mic, to get situated, and actually hear them. It's like a little trick which gives you an edge. It also helps when the front people are hogging the monitors.
If you're lucky (sort of) to be in a band with average singers, you won't be afraid of bringing down the level of the entire project when you sing. They might even encourage you through the embarrassing stages.
If you are contemplating taking the plunge, good luck! It can be fun, and you might get more work!