Yes, anyone can learn how to sing. When I first tried out for my band as a singer, I had literally no real experience. I could sing along with my favorite songs, and I figured it would be easy to do. Well, its not. I have recordings of my first tryout- I was tone-deaf, despite having a well trained ear, the tonality of my voice was horrendous, it was like an dying elephant with a cold. Now, I'm not saying that I'm a great singer now, or even a good one. But, I found my niche, and found MY voice. After trying to emulate my favorite singers for years, I realized that I'll never be able to be Phil Anselmo, or Maynard J. Keenan, or Neil Fallon (Most underrated singer/lyricist ever), and the quicker you realize that you're stuck with YOUR voice, the sooner you can improve your tonality and find what works for you. Not everyone is meant to be
the frontman, and quality backup singing is hard to come by. I have a friend who likes to sing, but he sounds terrible by himself, even he admits it. But, when we do some harmonization together, his super nasally voice coupled with my throaty, scratchy voice actually sounds quite nice.
Look at Les Claypool from Primus. Hes got an ultra-nasally, sort of annoying voice. I bet when he started to sing at first he sounded ridiculous, but, he developed his voice to work for HIS sound. Now he's known not just for his bass chops, but also as one of the most 'original' voices of modern rock.
So basically, if you want to sing, there are lots of ways to get better (as you have read in this thread) but the most important thing is to find where YOUR voice will work.
Also, one last thing. You can take lessons, do vocal exercises, practice, drink tea before a show, yadda yadda....
But the best teacher is EXPERIENCE.
Even if you think you suck, even if everyone who just saw you think you suck, most people will be encouraging and tell you how good you sound. If they're actually honest (a rare trait) they'll tell you where you need improvement. So, you'll get a boost of confidence (Which is sooooo important for performers) and if you're lucky, you'll get a truthful critique and you can take it or leave it.
Then again, maybe you just need to get surgery on your vocal chords.