Okay, if you're started to sharing your favorites, than let me participate!
I'll list my favorite characters (not all of caurse because I like really a lot characters, but that few that I like most) and reasons why I like them. May be it's slightly superficially, but it's my opinions how it is. Let's go (I'll add arts for some of them).
(P.S: idk, where arts were originally posted I've just used search by image so please don't hit me with your slippers)
Saber - Siegried
I love classic fantasy for that image of medival/feodal Europe (maybe not realistic but cool), so I love knights like theme and also I love dragons. Siegfried is a both, why I shouldn't like him?! (link)
Archer - Gilgamesh
It's hard to say why exectly I like him, but after watching Fate/Zero he became one of my favorite characters and... according to his popularity not only I think so. So there is something catchy about him. (link)
Lancer - Scathach
I like how she looks like but in general case this is not enough (with one exception, but... this exception is not exception from general rule cause the other points is also checked about her) for me to like character, as I alrady said some times here I come here after Fate/Zero and I don't look on F/GO like on game where you just collect your waifus (but still I'm also man not the perfect (or nor perfect, idk what you need to be prefect) being, so yeah, there is still few character that I like for that reason), I come here for that things that I find in that anime. So... for dark epic fantasy with many heroes from namy mythologies or realy history, but may be in a slightly strange interpretation. And with Scathach it's the case, cause she is not like one more gernder swap from original character and not the character who wasn't important in her story. So that's the case I simply like her. (link)
Rider - Constantine XI
I already sayed for what I come here... so Constantine XI is a prime example of whay I come for. Despite all the historical innacurate that devs leave in this university to make it interesting his armor or appreance doesn't look too weird or something. And also I just like how they handle historical background. He fall with his walls in the real story, but he won't let it happen again. That is my point and it's why I like him as character. (link)
Caster - Merlin, Merlin (Prototype), Solomon (True) (I shoulded delete his art cause it was deleted on the source, sad :()
Just like in the case with Constantine I just like how they processed the sources. (link)
The same is about Merlin (Protype) except her being genderswapped and being in my taste (read as "waifu") as like Scathach. (link)
Assassin - First Hassan, Semiramis, Kato Danzo (don't have an art)
Also same as with Merlin and other, but I like how they adapted history and myth to create this character. So yeah. (link)
She is that exception about what I was talking in the part about Scathach. I can say a lot about her... but I'll be short. In general she is just completely in my taste. I some her particular qualities may be subjectively better, but if we put it all together she is just closed to the prefection from the aesthetic point of view. Also she has a dragon. About her character... I can say it could be better cause she behaves herself well only with Amakusa whom I don't like. (link)
About Kato I can't say a lot, guess, but... there is not a lot of assassins who look like they are shown in popular culture (also this is the reason why it's hard for me to find assassin that I'll like despite having atleast two (if not three) of my favorite charactes in this class) and she is subjectively best of them.
And the last but not the leas - berseker - Lancelot (did anyone doubt it?)
He is my the most favorite character from all the list. First of all I just liked him cause as I already said I like knights theme and cause he was wearing just full plate and how it looks like. I just like his overal desing very much. Also I like his story despite it don't being shown directly (I mean like main story of any product) it's still tragically enough so it is. He also might by my favorite saber, but... his design as saber class servant is awful (imho). If only his saber verions looks like on the second art with him that I placed here... but we have what we have. (links 1 2)
About extra class - then my favorite is Merlin (Prototype), but I already put her in caster section cause she was caster in arcade.
P.S: sorry for my english.
P.S.S: there is also a few character that I want to say about despite not being my favs. There is just that case where the name of the character changes everything.
Quetzalcoatl - her overal desing looks not bad. I can't say that I like her a lot, but... can't say that I don't like her. But... while they saying that she is Quetzalcoatl I'll don't like her just cause she could be much cooler is she was a rainbow dragon (did I already sayed that that I LOVE dragons?)! And instead of it we got just girl in indian style with not bad, but not very cool design. I don't want to offend her fans, but that's my point (also the same with Tiamat).
Dobrynya Nikitich - almost the same. On one hand I should be glad that they finally started to adding characters from Kievan Rus' and I can see familiar names. But on other hand... I just can't image her prototype looks like that! It's too radical a change! But, still if we forgot about the name... she looks cool. Both girl and a dragon, so idk what to think about her. That's all what I wanted to say. Thanks for reading. Idk why I have written this.
