What is the best way to approach people in my network (or people I network with) to ask for job openings?

I think it helps to ask specific questions. as opposed to "I'm looking, can you help" try "I saw that there is a position listed on your company's website, is it still open?" or "I noticed on LinkedIn that you're connected to so and so, do you think they would be open to an informational interview?" If you don't ask something specific you get the issue people have when yelling "HELP!" in a crowd, you get a much better result screaming "FIRE!" Its better to be open about what you want upfront, if you email someone asking what they are up to, and then 5 emails later ask if they have a job for you, they'll realize that's all you were really after and feel used. Personally, I only ask direct questions (like above) to people I'm well associated with (especially those who know my work ethic). But you need to make a habit of becoming 'well associated' with a large network. Freely giving your time and advice is the best way to get something in return.

I agree with Pamela D'Luhy that you should ask specific questions, but you should also have specific requests.  During my job search, the best advice I received was to create a Networking Brief document.  It is awkward to walk up to someone and hand them a resume, but the networking brief is a one-page summary of who you are, what your skills are, what type of position you are looking for, what type of company you are looking for and what individuals can do to help you.   Remember that if people are even meeting with you, they are aware you are going to be asking for help, and they do wnat to help you, or they wouldn't have met with you in the first place. Make it easy for them to help you by using something like a networking brief.  Send me a message if you would like and I am happy to share with you the document that I used.   Finally, ask permission to follow up with every person you speak to. Add them to an e-mail list and send monthly or as apporpriate updates to your search and include your networking brief. How people can help you best might change and you want those poeple who have agreed to be an advocate for you to be on top of those changes.   Good luck!

I'd suggest focusing on talking about what opportunities they see or interesting things that are going on, opposed to talking about what you want. Focus on what is going on that you could engage with, versus just your need or desire.

You can use social networks like Linkedin which have premium subscriptions. In India we have more social networks like http://www.Buddyref.net (buddyref) which help people with job referrals.