I think all of us had at least once a character in a Drama which we loved more than the lead or even the second lead characters. So which one is your favorite and why?
For example:
Mo Il Hwa (Inspiring Generation):
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He has something mysterious and a lot of dignity. I couldn't predict what he is doing next, or even if he is good or evil but no matter which side he takes he is honest all the time even with his enemies
Lee Jin Sook and Soo from Cruel City
Jamie from Liar Game
Mainly 2nd leads... but they deserved better!

Cheese in The Trap: Baek In Ho, Baek In Ha & Kwon Eun Taek
Goblin: Grim Reaper & Sunny
Heartstings: Yeo Joon Hee
The Bride of Habaek: Bi Ryeom
I'm obsessed with Jung Hyun-Soo from Suspicious Partner, but Ji Eun-Hyuk (lawyer friend) was also really cool. Jong Hyun Soo was such an intriguing, dark character. 
Second Grim Reaper from Goblin. Awesome writing for a secondary character. 
Adjumma from Healer! Simply loved her character. Was funny, independent and super smart. Very loyal even though she pretended to not care. More characters like her, please!
Kim Byung-Yeon from Love in the Moonlight (the guard/best friend). My god, he was perfect in every way. Strong, mysterious, caught in his own situation, but still as loyal as possible. 
I loved Wendy from Falling for Innocence. Great comic relief.
Yoon Myeong-Joo from Descendants of the Sun was one of the best female characters ever. Loved her character. She was strong, level headed, brave, had great work ethic and knew what she wanted. Never meek or whiny. What an amazing role model.
Cheese in the trap :Beak in ho 
High end crush : section chief heo
Cinderella and the four knights: kang SEO woo
sassy go go: ji soo's character 
hi school love on : sungyeol
All supporting characters from Reply 1988. Each one had their own charm and they were wonderful neighbors/friends.
Jin Goo (Seo Dae Yeong) from DOTS. Made me love the 2nd lead couple more than the Beauty & The Boss couple.

 
Currently Dong Cheol in Save Me

Shin Woo (Yonghwa) in You're Beautiful, sure there were other really great side roles but for some reason this one just got to me and I can still remember the way he bought her the heels, and the "phone date" thing.
Adding to my list since I can't believe I forgot one, plus a new one I've found recently:

Gong Tae Kwang in Who Are You - School 2015
Dong Cheol in Save Me
Does Sunny and Grim Reaper count? they're more like 2nd leads to me than support.

That said, any supporting character played by Lee Kwang Soo and Kang Ki Young :D

I love these two lol
Dream High has a cute 2nd lead ship (IU x Wooyoung)
Sassy Go Go includes supporting characters' story arcs well
I agree with Linus about Suspicious Partner's supporting characters
Blood: Joo Hyun Woo
Hit the Top: Lee Won Bin [MC Drill] and maybe even MJ