Your various GPAs are not too much different (a good thing IMO), but don't list more than two, probably just use your overall 3.1 as long the position is preferred that applicants have a 3.2+. How 'core' or 'major' is defined the school varies but I'd expect most employers just think of 'major' as the courses in your area of study.
I'm still in school for engineering so have no authority on this, but I do think find it very surprising that people here would toss out any resumes that have GPAs listed when they are from a person still in school or have just graduated. For internships, a good GPA is very very helpful.
I plan to list my GPA for the career fair in a couple of weeks. I have some previous experience and project work, but not a whole lot, which is similar to a lot of peers, so a good GPA listed might get me an interview over two similar candidates. Sure there is people, and one person I know comes right to mind, that have a ~3.0 or less but have an amazing amount of project work and internships. These people would definitely get an interview before me, and be more likely to be hired unless they blew it or I amazed them.
I've gotten As or ABs in quite a few classes where at the end I can honestly say I know nothing about whatever topic the name of the class is, but that is often just the instructor. OTOH, a big reason I have a good GPA is I work hard, and I do not turn things in half ass; they are turned in when they are perfect (outside of a handful of things I just ran out of time on). Last semester, I sometimes wondered why I was spending 6+ hours outside of a 3 hour lab to finish it, when each lab was only worth about 2% of my total grade, while I had exams or other stuff to do that would affect my grade 10x as much. Except most of those extra hours was me perfecting the code... like I could've been like 90% of the class and just kind accepted that it sort of worked but damn did I spend the time necessary and get it 100% working, every lab. It felt awesome to have it working perfectly and to know everything about it after spending that much time.
So a good GPA can show that you are motivated and hard work to see things through. If I were to tell such a story during an interview when asked about my strengths, and it is the honest truth that I just sort of have a principle that I do things to the best of my ability, I think that that is one attribute that would be very important in considering to hire me, given my experience is not that great.