These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Base Breaker: Not so much the character, but his actions at the end of Brubaker's run on Winter Soldier, where Bucky decides that no further effort should be make to restore Natasha's lost memories of him, without asking her what she wants. Fans are split between hating this and ignoring it, and not minding because he did it out of love.
Creator's Pet: Subverted, if not averted outright. The Winter Soldier follows pretty much the exact pattern one would expect a Creator's Pet to follow: He comes in by killing the main character's biggest villain, gets a huge story focusing on him that reveals he has a deep connection with the hero, slowly takes over the main narrative and becomes the primary character of the story, gets retconned into another big character's backstory as the one responsible for their badassery before they become their love interest, and all the while gets to be really angsty and broody. However, fans adore Bucky all the same, primarily thanks to Ed Brubacker being just that good a writer. It goes to show that if you have a good writer, they can make anything turn out golden.
Genius Bonus: His codename. While the in-story explanation is that he was kept on ice between missions, it also references the "Summer Soldier" described by Thomas Paine as one who shrinks from service in the time of hardship. The Winter Soldier is thus one who endures.
Harsher in Hindsight: When Steve came back to life, Bucky assumed he would also return to being Captain America. We see him in his loft designing a new costume and identity, because he "doesn't want to go back to the whole Winter Soldier thing". Fast-forward to 2012, and sure enough, he's in a series literally titled Winter Soldier.
Harsher still, in All-New Invaders, he tells Namor and Jim that his name is Winter Soldier now, and "time we all got used to it."
Iron Woobie: Bucky was orphaned at a young age, fought in one of the worst wars mankind has seen, lost limbs and his memory, was made to betray his country and act as an assassin for sixty years. When he began a relationship with a young Natalia Romanova, he got put in a jar. He eventually got his memories back and he saw his best friend killed. It got better after that, until he was hung out to die by his government, convicted for treason, extradited to Russia , and eventually faked his death.
Replacement Scrappy: Averted. Bucky's run as Captain America was so beloved that even after Steve's return, the fans still wanted a little bit more of him. Its somewhat notable that it seems expected he'll take over as Cap in the films, and pretty much all interviews with his actor, Sebastian Stan, tend to talk about the possibility of him doing so.