Arc Fatigue: In season 2 of the anime, Mikoto's side of the Sister's arc takes 15 episodes (Episode 2 to Episode 16). While it is widely agreed to be better than the Index version of the same arc in nearly every way, the fact remains that the majority of the audience already knew exactly what was coming, so it was a bit frustrating it was dragged out so long.
Thanks to the Railgun anime, Saten Ruiko has become one of these. Her fans find her a refreshing character among the Four Girl Ensemble due to being an Audience Surrogate with no powers, to the point where she was popular enough to make it to the Top 8 during Saimoe 2010. Her detractors call her a blatant case of This Looks Like a Job for Aquaman, being put in plot-relevant situations to look good at the expense of the competence of the other main characters no matter how little sense it makes.
Haru'ue is not the most popular character, but she's pretty well-liked in the community. While she's not quite hated, she has a bigger role in the second half of the first season than most people would prefer, largely because she was basically created just as a device to connect Kiyama's children back to the plot.
Broken Base: The anime original arc that ended season 2 has caused a lot of division among fans. Many finding the camp and Power of Friendship levels over the top, being angered at how Misaka was given the Idiot Ball a couple times, and expressing frustration at the many plot holes in the arc. Others absolutely loved the arc since it addressed the primary problems of the Sister's arc (notably Misaka not going in alone) and everyone getting a chance to shine against STUDY.
Broken Aesop: During the Daihaseisai School Festival, Tokiwadai faces off against a school made up of low level ESPE Rs and Level-0s in Balloon Hunter, an event where each team attempt to pop balloon located on the heads of the enemy team by throwing balls, removing them from the game. Tokiwadai assumes an easy victory and thus does not form a plan or work together, while the opposing school already had plans and used teamwork allowing them to overcome Tokiwadai. This suggests an attempt to demonstrate how people without power can still achieve things through effort. This doesn't work very well because the methods used to achieve victory involved ridiculous rule abuse and relied on their detailed info on every member of the opposing team, such as using a fake spider to incapacitate the arachnophobic student and since powers could only be used on the balls not people, they did not throw them and instead tried to pop the balloons by hitting them with the ball still in their hand, which was allowed despite the rule explanation using the word throw. Lastly, the student who developed the plans was a criminal who specialized in finding ESPER weaknesses and developing counters and who gained the information needed for the plans from illegal sources, completely destroying any indication that people without superpowers stand a chance in a fair match.
Kihara Gensei, the patriarch of the Kihara family, is responsible for nasty experiments on children, which resulted in many of them suffering permanent trauma, all so that he could create a powerful Level 6 Esper. Gensei has no compunction murdering people in his path. The first "sample" of his experiments was his own granddaughter: Gensei overloaded her mind to collect samples from her, and repeated this process on other children as well. Gensei was fond of manipulating his own family members into becoming his test subjects, even though he knew that a massive surge would possibly destroy all of Academy City.
His granddaughterTherestina Lifeline is just as cruel and twisted. Wanting to follow Gensei's example, Therestina abducted children and put them in comas so she could harvest the samples from their minds. Gensei manipulated the young heroines into providing cover for her, all the while intending to overload the minds of the children she kept under sedation, fully aware this would destroy all of Academy City and not caring as long as she achieved the "perfection" she hungered for.
Draco in Leather Pants: Misaki Shokuhou. This is probably the only reason why she appears right from the start in Railgun S.
A one sided crush between Miho to Saten and a not so appropriate obsessiveappreciation from Kuroko to Mikoto. Also, Saten's trademark inspection of Uiharu's panties.
In the second half of the second season, Saten and Uiharu essentially adopt a child together.
Due to the way Kinuho and Maayanote Kongou's posse fight, it's hard to not see them as a Battle Couple. They even have intimate eye contact throughout their battle.
Mikoto had a dream where she was Misaki's maid and had to do things like give her a massage. It is implied Misaki had the same dream, but in reverse.
I Knew It: We all knew it when Telestina was revealed as the villain. But what caught everyone off guard was the revelation that she is Gensei Kihara's granddaughter. Fan reaction about this between the first time viewers and novel readers were varied. With the former going "What the? She's his granddaughter?" while the later going "Wait, She's a Kihara!? As in the Kihara's?Oh Shit!?"
Moe: Kazari, for certain. Ruiko can also count as this; as mentioned above, she made it to the Top 8 during Saimoe 2010.
For Gensei Kihara, having 10 kids forcefully overloaded with esper powers definitely crosses the line. And before that, using his own granddaughters as the subject of his experiments. One was involved with the Level 6 overload experiments, the other became a cyborg!
Therestina Lifeline Kihara planning to overload the child Errors that her grandfather created so she can become god and destroy the entirety of Academy City to boot. On top of maliciously manipulating the cast, practically turning Uiharu and Kuroko against each other, and her brutal beatdown of Misaka after she used her Kryptonite Factor on her.
While Index was a hit, Railgun was more popular, sold more DVDs and got a better reception with fans thanks to a lot of Character Development for the cast. Note that this is only true of the Index anime. The Index novels sell hundreds of thousands of copies and are the most popular medium of the entire verse by a healthy margin.
Probably because it's also partially a Schoolgirl Series. Furthermore, in a world where crazy people wields crazy superpowers that get thrown around with reckless abandon, the main characters of Railgun seem like relatable humans (see the case with Saten, above).
Padding: The anime increases the importance of Saten and Uiharu, adds a fifth girl from a filler arc and then has the three of them, Kuroko and Mikoto frequently wandering around making small talk and eating things. Not much of this is really relevant to the ongoing plot arc, so when an arc gets into full swing, the three less important girls tend to get brushed aside so that the original mood of the story can remain intact.
Because of her actions at the time, Uiharu was absolutely loathed by the fanbase throughout most of the final episodes of the first Railgun anime season, it was only when Kuroko gave her a much needed slap-in-the-face that people started warming up to her again. It also helps that this took place in anime-original content; giving Uiharu a free pass despite her actions.