Off the top of my head, I can say I actually kind of enjoy the trope of the character who is adamantly chasing after the protagonist. The character has some (typically) misguided views that the protagonist is a scummy criminal and needs to be brought to justice, but (usually) ends up seeing the wrong in their own stubborn ways by the end of the series. Examples include Inspector Javert from Les Miserables, Smoker from One Piece, Inspector Lunge from Monster, etc. Also a more silly version of the trope is Inspector Zenigata from Lupin the Third.
The kind of trope I really dislike? The 'master' of a villain trope... I've only seen this example in anime/manga and will just list who I'm thinking of to help paint the picture: Babadi from Dragon Ball Z and Kira's father from Jojo's Bizarre Adventure: Diamond is Unbreakable. What makes this type of character so annoying? They sit on the sidelines and very annoyingly overreact to any danger (no matter how little it really is) to the villain. Kira's father is even more annoying, because he floats around in a picture, blabbering his mouth really loudly where the heroes should really be able to hear him, but no, he just floats away and comes back another episode.
Also, I really don't care for protagonists who are just overpowered and... that's it. Like the protagonists from Seven Deadly Sins and Sword Art Online. I hate the SDS protagonist even more because he constantly gropes the female protagonist, but that's a complain for another topic, as it's not really a character issue, but a more overall issue.