I've been singing and playing bass for probably close to 25 years now, so it's virtually second nature to me. Of course there are some bass lines that I still find it impossible to sing over, but there are a few bits of advice I'd give that I think have helped me to pretty much be able to sing and play most lines I need to without much extra effort.
1) Learn to play drums. Not only invaluable when it comes to timekeeping, but because you're conditioning your body and your brain to do so many different, often disparate things (if that makes sense), I've found it really helps when it comes to singing and playing. Oh, and by the way, I also used to sing and play drums at the same time.
2) Learn and memorise the lyrics to the song first, before attempting the bass line. I find that not having to really think about the lyrics is a massive boon. After all, the bass is the hard part to pull off, realistically speaking. Singing is just talking in different pitches really.
3) Bit hard to explain this, but separate the bass line and the melody of the song in your head, before you attempt to play it. Like I say, hard to explain but it works for me.
4) Don't be slavish to the bass line unless you really have to be. If needs be, simplify it where you need to. Unless you're in a tribute band, you don't need to play a bass line note for note or verbatim (bar something really specific and integral to the song... a good example being something like Day Tripper by the Beatles) so, if you can, dumb it down. Someone cited Spirits in the Material World by the Police earlier in this thread. Great bass line, but listen to what Sting plays when he does it live. It's drastically trimmed so it's probably about half as busy as the studio version.
5) Play some air bass! Sounds stupid, but before you even play a note on the bass, see if you can co-ordinate your theoretical finger movements for the bass line while singing. Or put more simply, try and replicate the rhythm of the bass line with your hands while vocalising. If you can do that, actually playing the bass and singing should be no issue.
Horses for courses, these tips won't work for everyone. But they've worked for me, and even to this day if there's a fairly complex line I need to sing over, I still return to some of them.
Hope this is of some use to someone out there!