I sing some harmony in the country band I play in. I'm not as comfortable with it again yet as I was when I played with a different line-up of this band a few years ago. Back then, I had never ever tried to sing harmony, but after working some parts out with the lead singer, I got pretty good at it, and eventually could usually find a part. I didn't (and still don't) usually ever hear the original version of these songs, not being much of a country listener, so I just figure out my own bass lines and harmony parts, unless the singer shows me a harmony he wants. The original version of the band broke up and so for about 4 or 5 years I didn't sing at all. We got back together last year and I started singing again. It was pretty rough at first, especially since the new guitarist also sings harmony, and we haven't yet gotten to work out three part things. So generally, if he's singing, I don't. And vice versa. I think we'll get some three-part stuff worked out soon, once we can actually arrange a full practice (the drummer and I live 25 miles from the lead guitarist, and 60 miles from the other 2 guys... Colorado!).
I can totally relate to the other comments about playing bass and trying to sing harmony. It's kind of tough to play bass and sing ANYTHING at first, since you're so tuned in to the root and 5th. And I have a fairly high voice, so that makes it even tougher to find a part when half of my brain is concentrating on LOW notes! I enjoy it a lot though, and want to get better and more comfortable at it. I would love to get to the point where I can just harmonize on anything on the fly. That would be great for jam sessions and things.
Oh! One thing that helps a lot... when I played with the first line-up, I found that if I listened to The Beatles a LOT and took turns singing their parts... "Okay, now I'll listen to that and sing PAUL'S part this time...". One great thing about them is that they generally harmonized on the entire songs rather than just choruses.