Each hosting package comes with the Search Engine Submission feature in the Toolbox section.

How can I get a listing of what search engines, directories, and data supplies that Network Solutions submits the domain to and is this a onetime push?

Do you still recommend this submission for domains that have already been indexed by all the major search engines.

I'm trying to get a feel for what this feature really does.

Charles Self
Narrow Gate Solutions
www.narrowgatesolutions.com
dotnetnuke.narrowgatesolutions.com - DNN Related Resources

I'll try to get a list of the search engines to which it submits for you.
This tool will submit your site to search engines once per month, every month.
If your domain is already indexed, you should still submit, but do not submit with the tool if you are currently submitting the domain with some other tool on a monthly basis. A domain should only be submitted once or twice a month max.