To me I don't think it's that big a deal to ship real people as long as it doesn't go overboard. Like Ushio said, it's just nice to see good looking people together. Their public personas are just that. We don't know much about their personal lives or their real personalities. We just know what they want to show us and so they become characters themselves.
It just becomes a problem when people become disillusioned into thinking it's real. Of course there's times I hope all that teasing means something more, but I don't go all batshit crazy on their personal instagrams going "____ IS REAL!!!" And threatening celebrities' actual bf/gf for ruining the ship. Just keep it to the imagination and it's all good.
Anyway back to original question, I haven't shipped real people in a while, but was really into AKB48 (TakaAcchan) and SNSD ships (Taeny, Taeri). Hard to ship Takamina and Acchan now since Acchan graduated. I don't follow AKB anymore but I'm sure there's still a lot of cute flirting going on in there. SNSD shippers got too crazy for me. The fanfic for SNSD was great though. Some really great writing in the fandom, but gotta wade through a lot of bad ones to find them. I think MomoTamai from Momoiro Clover Z is super cute.
Hmm what else... I just recently got into kpop band EXID from some tumblr posts talking about how gay Hani is, but I don't ship her with anyone. I think lately instead of having a specific ship within a band, I just end up shipping the idea that "so and so is a flaming homo".
If you guys have some ships please tell e I am on girl band spree today and I am looking on some pairs (dont judge I saw a tumblr post of a girl band acting over friendly and I was like "HOHO!! who are they!!?"
Haha I'm curious about the post and band XD