I'm surprised I haven't found this thread before. Here are my thoughts:
Honestly, for me, I think Shinji X Kaworu is one of the best ships especially since it's part of all the storylines for the franchise (Manga, OG Series, and the Rebuilds.) The two of them are adorable together and function as good opposites. If you want a good (but brief) analysis of the two as a couple I recommend watching GoatJesus's "Examining the Yaoi BL Genre", it's FANTASTIC how he analyzes them (I mean, he IS a hardcore Eva fan, after all, so...yeah.
) Dear lord though, the fanfictions/doujinshi stuff is just...YIKES!
Other couples would be Misato X Kaji (again for how good the two function as opposites to each other.) Shinji got to see it all that at the end of the series too. Also though it gives us more insight into why she's so sad, so not only is it cute, but it's also part of the main story as well like Shinji and Kaworu.
Shinji X Asuka (while cute), really doesn't seem likely. They seem to bond in Episode 15 (I think) with their kiss very situationally like Asuka just wanted to do it to kill time with it and nothing more, though if you read deeper, you could see it as her looking for a connection and her then being disappointed when doesn't find what she's looking for in Shinji. However, THAT is just a theory from GoatJesus and his Shinji Ikari Character Analysis video, again, check him out he's great (not trying to promote him on here or anything.)
Other than that, Ritsuko and Maya are cute together, after all, it seems Maya has some sort of lover-type attachment to her, she calls her in "senpai" in the Subs and her person she loves the most when the EoE Third Impact takes place is Ritsuko.
I've seen a lot of people who like the Asuka X Rei ship as well, so I guess that can be #5 too. They do contrast nicely as well in the show *cough* Elevator Scene *cough.*
Yeah, I'm probably missing some bigs ones off people's list, but these 5 are just my personal favorites.
"Eva broke me so hard...I fell in love with it HARDER. Now I think Shinji x Kaworu is the hottest couple in all of Eva...yeah nothing wrong here." -This Boi
(Youtube: Nario, the one with the Ness Picture. Can't link it on my profile because it's broken.)