Every strength has the potential to become a weakness under the right circumstances. If your character has an enhanced ability, don't just consider the ways that that ability could help them, also consider all the ways that that ability could become a hindrance or be turned against them by their enemies. That one-shot magic beam of awesome your character can make should probably drain the character's energy dry when used, as an example.
• Also, in terms of skills, one thing that you can do to bestow more realism to a character and edge them away from sue territory is give them fringe powers and weaknesses as well. Let's say the character has an unnaturally or magically high empathy. One fringe benefit is they may become an Omniglot and be able to speak to anybody in their native tongue. The issue could be that they can't fight because it hurts them just as much as the enemy. Also remember to build around the ability's weaknesses and strength as well. The character might be illiterate in the setting because nobody uses the writing system they use, because his ability only allows him to communicate to the person they are talking to, also making them speak in a foreign language when not directly addressing people. Maybe this character is wonderful at dodging because they can sense their attacker's next move. Give them that depth that makes each ability not only useful and harmful, but also how it branches out to be both good and bad, because a sue whose abilities are only "harmful" to them as well as only helpful is still a sue.
• Don't talk about skills that your character has that aren't important to the story just because you think they are cool. Even if they don't show up in the story, they are meant to help flesh out a character, not to make her look better.
• To expand slightly: Cake-baking skills in a story about a cake-baking competition fit. Cake-baking skills that are used to cheer up another character or poison an enemy (or a friend, accidentally) also fit. Cake-baking skills in a story without ovens or cake do not fit.
• Don't make your character better at everything the main characters can do, because then you don't have a character. You have The Munchkin. If you have to, choose one skill from a main character that your character is better at, and make sure this character has a good reason to be more skilled. Let's say one of the main characters is a swordfighter. If this character has only been training for a few years, and your character has been training since childhood, you have a good reason. But this doesn't mean your character replaces that main character, only that your character has more experience in that skill. If used right, this can actually be beneficial to the character; giving him/her somebody to act as The Rival to them can kickstart character development, if capitalized upon.
• Also realize that the higher up you go in education, sports, or any creative field, the more you will hear the saying "talent is cheap". To be the best of the best you need to literally dedicate yourself to something body and soul, and you need to be working at it every single day. There is a reason not everyone is immensely skilled at even one thing, never mind the two handfuls The Munchkin is going for.
• Few abilities translate as well as you'd think. Driving a car doesn't mean you can fly a plane. It doesn't even mean you can drive a motorcycle. Fencing with a foil doesn't mean you can engage in combat with a katana. Michael Jordan was an awesome basketball player, but he kind of sucked at baseball. One skill may aid in learning another, but it doesn't equal it. Knowing how to play the piano means you can read sheet music and keep time. It doesn't mean you can pick up a violin and turn out a concerto.
• Make sure that the skills your character has fit the world she lives in. A medieval girl who knows how to play a flute is plausible. A medieval girl who knows how to play a magical electric guitar, African drums or the didgeridoo is not. Similarly, if your character has no real reason to know a skill, don't give it to her; for example, resist the urge to give your thoroughly non-Japanese character a katana, even if Katanas Are Just Better. Stretching the story to fit in a convenient sword master will not help.
• As a corollary to this, don't give your character skills that his or her gender, social class, race, age, species, physical ability, culture, what have you, would prevent him or her from having, unless you can give a really good reason for why he or she would have that skill. For instance, an upper-class girl in the 18th century would most likely not know how to command or sail large ships. Also, if canon characters with similar limitations couldn't do it/learn it, your character can't.
• Note that if the story involves Character A traveling in time to meet Character B, and they bring along said magic instrument, than make sure Character B is confused about it. Character B likely hasn't seen anything like what Character A's magic instrument is, so make sure to have the traveller explain it to them.
• Similarly, keep an eye on the skills of the characters and respect the responsibilities they can take because of their skills. If a character is Too Dumb to Live it just comes over as coincidental that he suddenly managed to achieve things that are normally reserved for people with intelligence in that field. Likewise it will come off as unsympathetic if a character is very intelligent but uses none of the intelligence given to it.
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How to not write a Mary Sue
RandomA Mary Sue is an idealized and seemingly perfect fictional character. These characters are often irritating and cringe worthy. This book will tell everything about identifying a Mary Sue and how to avoid them.