I did notice the backlinks showing in Google Webmaster Tools for about 1 week a month or 2 ago. Then they disappeared. I had not made any changes so I thought they would appear again. They have not. And they do not show on any back link checkers.
The .net is only 3 months old so not ranking well. It has about 30 backlinks itself but there should be 10 from the redirect from the .org too.
When you refer to the “header” what do you mean? The 301 redirect goes to the .net so no content is on the .org domain name. It is a permeant 301 with GoDaddy.
Did the .org rank well previously, and did you follow Google’s “change of address” guidelines?
I’m not convinced that 10 backlinks are going to make very much difference (unless they were from exceptional sites), but have you checked that the links still exist and still work?
Immediately after the site move is started, try to update as many incoming links as possible. These include:
External links: Try to contact the sites in the saved list of sites linking to your current content, asking them to update their links to your new site.
Profile links such as from Google+, Facebook, Twitter, and LinkedIn.
Ad campaigns to point to the new landing pages.
That suggests to me that they don’t list redirected links as backlinks (which would make sense; they’re not actually links to your current domain).
If the links you’re worrying about are those types of links, which you’ve placed yourself, then I’d advise you to forget about them. Google places very little weight on those kinds of links (and advises site owners to make user-submitted links as nofollow).