I lived in LA until college, and am now returning... West Hollywood is a great area for gay and straight alike. I would say don't live there if you are homophobic, but it's a wonderful neighborhood. If you're working in West Hollywood, you could easily look at these other neighborhoods depending on your budget:
Hollywood Hills (esp Beachwood Canyon if you can afford it)
Los Feliz (north of Hollywood Blvd).
Silverlake (north of Sunset Blvd)
Parts of Mid-Wilshire (Miracle Mile, Hancock Park, Carthay are all nice)
I'd stay out of Koreatown, Echo Park, Downtown LA, etc. Some more good resources are:http://www.lapdcrimemaps.orghttp://en.wikipedia.org/wiki/Los_Angeles_neighborhoods
Note that none of these neighborhoods are especially cheap. If you want cheap housing south of the SF Valley, you have to live in a somewhat questionable neighborhood, settle for someplace tiny, or get really lucky. Craigslist is a great resource for market research, and is composed largely of small property management companies and individual landlords. If you're more the large apartment community type, http://www.apartments.com
and others like that list most of the corporate apartment complexes.