It's hard to say. For instance, the bike path that goes under the 405; I won't ride that part because 4 years ago a guy got thrown off his bike and beat up there. Yet, I have a friend that rides it all the time. Go with your gut.
Here are a couple of good resources: http://la-bike.org/events, http://www.bicyclela.org/
Sorry I can't be more specific regarding areas.