- Useless (female) characters. By that I mean characters that don't seem to have any useful purpose and spend the entirety of the show either acting as cheerleaders or getting in trouble and giving the protagonist a reason to fight. The reason why I single out females in particular is beacuse it is often female character that fulfill this role; Winry Rockbell from the original Fullmetal Alchemist anime and Anzu Mazaki/Tea Gardner from Yu-Gi-Oh! Duel Monsters being examples. However, this is by no means limited to female characters (Honda Hiroto/Tristan Taylor, also from Yu-Gi-Oh! is arguably even more useless).
I understand that not every character in a certain show can have a certain role, gift or ability, but that doesn't mean they should be relegated entirely to the sidelines or be made completely unimportant. Riza Hawkeye from Fullmetal Alchemist and Mirai Yashima from Mobile Suit Gundam are good examples of characters who don't fulfill any primary role, but still maintain a very good degree of relevancy to the overall plot. Basically, if you are going to include someone in the main cast, you should make their presence justified.
- The-not-so-useless-but-still-useless characters. Unlike completely useless characters, these actually are actually capable to a certain degree and are relevant to the plot....but that is not saying much. While they are capable enough to be a part of the main cast, they are incompetent enough to always get in the way and be a source of trouble. These characters are arguably even worse than the useless ones, since those are at least smart enough to be aware of their uselessness and usually keep themselves out of harm's way. Fa Yuiry from Mobile Suit (Double) Zeta Gundam is an example.
Don't get me wrong, I am not saying characters should never get in trouble and require help. But when this occurs so many times that they are practically a liability, you start questioning why the rest of the cast doesn't have the common sense to force them to safety or at least scold them for being a burden.
- Comic relief characters. To be fair, I despise comic relief characters in all mediums, but anime is by no means an exception. To have such a character is to admit that you are incapable of integrating humour in your show in a natural, more subtle way and need to condemn an entire character to uselessness in order to devote them to easing drama and tension. To me, comic relief characters are a sign of amateurish work and I cannot express how much disdain I have towards them.
There are plenty examples of this (Boss from Mazinger Z comes to mind), but I am going to mention Master Roshi from Dragonball Z as a particularly tragic example. While Master Roshi was very often a source of humour even in the original Dragonball, he was actually fairly important due to being a mentor for Goku and Krilin. After that, however, he is entirely relegated to being a source of humour. A particulary eery example is the Broly movie, where his practically entire contribution to it was a strange outburst he makes that causes Goku to look at him bizarrely for a moment before continuing his fight with Broly. Oh, how the mighty have fallen...
- Annoying fanservice. Unlike most people, I will be the first to admit that a bit of fanservice doesn't necessarilly hurt a show. But when it is done frequently and/or in a "in your face" manner, it starts to get annoying even when you are the intended audience....especially when you are the intended audience. I can't recall her name, but there is a female character in Bleach who's breasts are so enormous that you literally have trouble paying attention to the show.
While we are on the subject, Gainaxing is a particularly atrocious example of fan service. To clarify, I do allow for the fact that it might fit in certain situations for certain female characters; it particularly fits the devil-may-care types, since it goes along with their free-spirited nature (although it's still weird). But I find it double facepalm worty when a show has a female character that we are supposed to take seriously, only to see that she doesn't have the sensibility to wear a bra.
- Awfully impractical clothing. I won't even bother explaining this one, since chances are you have seen it plenty of times. Wearing high heels in extreme situations, clothes which reveal certain parts in spite of potential dangers etc. etc.
- Teenagers as protagonists. I don't mind the fact that teenagers are frequently protagonists...but do they really have to be all the time? There is nothing wrong with an adult or even an elderly perspective.