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.
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.
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.
Kihara Gensei is the patriarch of the Kihara family and a Mad Scientist responsible for nasty experiments on children, which resulted in many of them suffering permanent trauma, so he could create a powerful Level 6 Esper. Gensei has no compunction murdering people in his path in his mad obsession. 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 granddaughter Therestina 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.
Thanks to the Railgun anime, Saten Ruiko has become very popular with fans as she's a refreshing character among the Four Girl Ensemble, even more popular than Mikoto on some sites like the Nico Nico Douga community. In fact, Ruiko was popular enough to make it to the Top 8 during Saimoe 2010.
Don't forget Harumi too. She's pretty well-liked in the community.
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).
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.
Haru'ue is not the most popular character. 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.