I feel like I usually lean towards liking the flirts, or those characters I immediately think to dislike because of their attitude or something like that (though I get a bit curious about "why are they the way they are?"), but then am introduced to their backstory and realize they're very much in need of a hug; so I go from "I hate you" to "I hate everyone who hurt you, my poor baby". Also the ones that like to act tough but are soft and squishy on the inside (like a toasted marshmallow) are great too. c:
Deep underdogs. A character that has all the odds playing against him/her.
He/she is no a chosen one by the gods, or prophecy to save the world.
He/she is not destined to be married to some great king/queen/emperor of some kingdom.
He/she isn’t not born in some wealthy/powerful family.
They also tend to be losers and not rejected by society.
I love seeing those type characters struggle, working really hard with their own capacities and luck to overcome great obstacles. Saitama from one punch man might be a joke but he's close to that archetype I'm looking for. Gut from berserk is also a bit like that.
Naruto almost fooled me by making believe he was an underdog who had to work hard to reach his goal. In the end he became some sort of Jesus Christ who had a great prophecy, great family and had been entrusted with a grand destiny. The fact that he was also given a demon fox at birth predestined him to be stronger than most people around him.
Seriously this saviour/chosen one character archetype has been overused to death.
One who shows resilience. One who never gives up cuz he keeps getting up and fighting even if his ass gets constantly handed back to him over and over again and there's very slim chance he will turn this around and whoop the other dude's ass...
Goku... Naruto... Yusuke Urameshi... Kenshin... Ichigo...
Those guys lol
I really like the loner characters, most of the time. The ones that aren't very sociable, but still have a deeper insight into any given scenario because they can look at it from an outside perspective. There's often an aura of mystery surrounding them, which makes you want to get to know more about them, but their stand-offish-ness makes it hard for anyone to get close. But that just makes those moments all the more worth it.
First example that comes to mind is good ol' Carmen Sandiego. Mysterious, AND classy!
Of course, let's not forget the grumpy loner either.
I think another thing that I find endearing about the loner archetype is that they often have the funniest lines. Because they don't speak very often, the writers wait for the perfect opportunity to give them a line to solidify the punchline.
So, yeah. That's my two cents.
i really like characters that are a little silly and oblivious! you know, the kind that does or say stupid things but complete seriously and with zero self awareness of how dumb they sound. i also like characters with personalities that dont fit their looks (i think its called moe gap?), like that cliched shit where the gangster is actually sweet and loves cakes IM SORRY I SUCK BUT THAT IS MY WEAKNESS.
dork villains dork villains dork villains dork villains dork villains dork villains 2
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I LOVE DORK VILLAINS
it's a bonus if they're actually REALLY FUCKING POWERFUL but are too dumb to live up to their full potential of villainy. Lord Hater (literally my all time favorite fictional character) is a prime example.
(I also have a major thing for any character related to skeletons if most of the above examples didn't make that obvious enough)
Nerds. I like nerds. Bonus points if nerd has glasses, but they are purely for aesthetics.
Paladin-type characters. Yes, I like old-fashionated good moral characters. Sad you have to dig deep to find one nowadays.
Silent ones. Especially, if it's a visual medium - yes please!
Realist/optimist mix.
Hard-working type.
REALLY smart characters who aren't nerds. Bonus points if they ARE friends with a nerd.
Also, @Neonmeres He's a main character in shounen. No way in 7 Hell dominions he could be an actual underdog. But if you need one - ROCK LEE FOR THE WIN!
In short: I normally tend to like particular characters rather than types.
i like every character that actually plays an important place in the story and enhance the main character or gives it a purpose.
You know, like that wise figure, or the underdog who lives in the shadow of the main character but is so well fleshed out it makes you feel he has it's own path it's own destiny that just happennd to cross paths for a while.
If we are talking about personality, i like flawed characters that struggle with their own desires to achieve a ideal, you know romantic Heroes, and if they die it has to feel unfair, broken, but even then there's still hope because of the other characters.
My fave is definitely the hesitant type, the guy who didn't sign up for this BS but here we are and we're gonna push through it come hell or high water because there's something they need to do. This is usually how my headcanon for Link1 usually runs xD
Other than that, I love love love love love the cocky and stylish trickster type.
gawd P5 why are you makin all my dreams come true
Oh yeah I'm also a bit of a sucker for characters on redemption arcs
I love all types of characters, so I guess my favourite type of character is one that is interesting to watch, especially when dealing with hardships.
I especially like characters who have shit thrown at them 24/7 and deal with said shit in their own interesting ways. I'll throw in a few examples:

Guts from Berserk. His life is absolute turd from the very start, and he's surrounded by danger since birth, but the bastard just doesn't die. Many people in his position would have eventually broken down and given up, but he just keeps going, which I think is both interesting and awe-inspiring.

Kaneki Ken from Tokyo Ghoul is also a character that has to deal with a lot of shit. Because he's suddenly been placed in a position where he doesn't belong, he's forced to make drastic changes to his character, treading on the line between decency and unadulterated psychosis. He's an interesting watch, as he's constantly changing both in character and philosophical ideals.

And last but not least, Punpun from Oyasumi Punpun. Now, Punpun's hardships aren't as dramatic as the rest, but they're relatable due to its realistic setting and generally subtle tone. Punpun's hardships aren't immediate, but rather they build up and affect his psyche, which only breaks him down bit by bit until he turns into a complete sociopath.
I just realized that the characters I listed are borderline insane, so I guess that's my favourite type of character - the ones whose limits as the measly humans are pushed to, and maybe even past their boundaries, and watching them break down and build themselves up again.