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.
Base Breaker: Ryoma, Atobe, Horio, and to a lesser extreme Sakuno.
Best Known for the Fanservice: While many agree the series are not very well written, female fans admit they only read it because of the characters. This also is applied for the anime and, especially, the musicals.
Bishonen Jump Syndrome: By far, Prince of Tennis is the most infamous offender of this, since the entirety of the fandom are girls. In fact, most would probably laugh at the fact that this is actually shounen. I'm tempted to say that it's so much fangirl squee, that it can pass for Testosterone Brigade!
Several characters, but specifically: Sakuno (Ryoma/anyone shippers, but Momo/Ryo and Tzu/Ryo fans are specially bad), Yukimura (Atobe/Sanada fans) and Yanagi (Inui/Kaidoh fans). To a milder degree Momoshiro (Inu/Kai too), Kawamura and Saeki (Tezu/Fuji for both).
An doesn't have it THAT bad, but she still gets pot shots for "stringing Kamio along" and "playing with his feelings."
A lot of people like to forget that Atobe in particular tried to ruin Tezuka's entire tennis career by wrecking his arm in their match, for no real reason other than he wanted to. Sure he comes to regret it and has a moment of redemption, but it's a fact that seems to commonly slip from the memory of most.
Ryoma shaving Atobe's head while he was unconscious. While it was supposed to get a laugh at Atobe getting some Humble Pie, many of the readers actually found it reprehensible coming from Ryoma, The Hero of the story.
Koharu and Yuuji's gay jokes on court to throw Momoshiro and Kaidoh off, which a significant part of the fandom took as a very offensive take of homosexual men instead of the intended parody of the FanserviceHo Yay.
Hyotei. The entire team is the most popular rival school, to the point that Konomi managed to bring Hyotei back for the nationals. It's debatable if the fanbase is larger than Seigaku's. Special mention to Atobe and Oshitari, clearly the two most popular from the team.
Almost as popular as Hyotei, there's also the Rikkaidai entire team. The eight of them HAVE to appear somehow in the series. Mainly Sanada, Yukimura and Kirihara, and close call to Niou as well.
Another big example of this showcase is to count the returning characters for New Prince of Tennis. If a character is in that training camp, there you have your Fan Favorites, including the return of former Seigaku Captain, Yamato Juudai, and Canon Foreigner Ryoga Echizen.
From New Prince of Tennis, Irie Kanata is the only highschooler in the ranking poll, helped a lot by the fact that his voice actor is Hiroki Aiba (the thirth Fuji in the Tenipuri Musicals). Also it seems Tokugawa is shaping into another Dark Horse.
Ryoma is very frequently accused of being one of these by his anti-fans. When a 12 year old boy is already a 4 time tennis champ (in the US) it's kinda hard not to see it. But still, lots of the accusations come from rabid Atobe and/or Rikkaidai fans... who then go and make themMary Sues and try to pass them as Better Than Canon.
Other character who have been given the God-Mode Sue ball at one point or another are Yukimura, Fuji, Tezuka, Sanada, Atobe and Kevin Smith from the anime.
Also there's Ryoga, who counts as a straight up Marty Stu: He is Ryoma's older brother (it's explained he is adopted, because his mother died and Nanjiro took him in), he is better than Ryoma at Tennis and his first appearance in the movie showcases him doing awesome stuff (was hired by a wealthy man to play tennis on his team, played tennis against Ryoma on a sinking ship, and not only survived, but he also got away on a jetski to the sunset). His popularity was so big, that not only he appeared again in the OVAs, but Konomi also made him a Canon Immigrant by including him in New Prince of Tennis.
It's the Same, Now It Sucks: A considerable amount of fans and casual viewers seem to think this about New Prince of Tennis, since it's still the same plot from the original series with some aesthetic changes and a lot more new characters.
From the same portion, some people also think it's So Bad, It's Good, since it also feels like the series is now self-aware of it's own tropes, without taking itself so seriously.
Launcher of a Thousand Ships: ...Who has NOT been paired up with either Keigo Atobe, Tezuka Kunimitsu and Shuusuke Fuji in fanfiction and doujinshi, people? Sakuno also verges on the trope, since she's often used as a Possession Sue by some fangirls.
Also Ryoma, naturally due to being the main character and by far the prominent shota of the series.
Like You Would Really Do It: In New Prince of Tennis, half of the middleschoolers are eliminated from the U-17 Camp (among them there're Sanada, Niou, Inui, Kaidoh, Momoshiro, etc. and even Ryoma and Kintaro). You knew they would endure the Training from Hell and return to the Camp somehow!
From the anime, Professor Hanamura Aoi, from Jousei Shounan. She's creepily attached to her tennis team and is constantly fawning over them and Ryoma.
Also Konjiki Koharu, who aside from doing gay jokes with his doubles partner Hitouji Yuuji, is constantly fawning over Kaidoh or other hot guys. In the Pair Puri omakes, he has been seen having crushes on Atobe, Tezuka, Kamio, Saeki and Sengoku.
Misaimed Fandom: An Tachibana's actions during the Senbatsu arc were not supposed to be seen as "awesome" and "proper of a strong female character". They were supposed to be a part of An Aesop in regards to how seeking violent resolution to a conflict is not worth it. Specially considering that Kirihara could have died after that Staircase Tumble—and that almost everyone blamed Kamio for the incident! And that An herself broke down crying in regret when Ryoma unmasked her and Kamio called her out!
Moe: Sakuno and Tomoka, specially during the Jyousei Shounan anime arc.
Shipping Goggles: In many fics and for most of the female viewers, there're very few that don't get paired with another male character, and is more common to find fics where Everyone Is Gay.
The most common example are Seigaku fics, with Momo/Ryoma, Tezuka/Fuji, Oishi/Eiji, Inui/Kaidoh and Kawamura is het or paired with Akutsu, or any other combination aside from the Fan Preferred Couples. The same applies with Hyotei and Rikkai... or any other school, for that matter.
Ships That Pass in the Night: Happens a lot in fanfiction, given that there are A LOT of pretty boys to pair up, even if they haven't have canon interactions. With that said, that didn't stop Konomi from making all those omakes...
There're also other fans who ship the coaches Hanamura and Sakaki.
Take That, Scrappy!: For the ones who hated Ryoma, Tezuka bitchslapping him in the anime filler so he wouldn't play against Kevin is definetely this.
The Untwist: For some it was fairly obvious, since the latest chapter where it's revealed that Tezuka would be playing for Team Germany, right after it was revealed that Ryoma could switch to Team America just by location.
Rikkai fans have been whining for several years about how the team was portrayed in the anime and about The Switch between Yagyuu and Niou not being featured in their match. Lots of fans of the manga often say it of the original after the Hyoutei arc (the anime had caught up to the manga, changes started to be made).
Subverted later, when the OVA's start covering the Nationals arc and get praise for portraying some rather... polemic manga scenes in a much more understandable and sympathetic light. I.e., Ryoma cutting Atobe's hair when he's still almost unconscious in the manga is switched to Ryoma having second thoughts about it, Taki offering to have his own cut hair instead to spare Atobe the humiliation... and then Atobe swiping Ryoma's razor and willingly cutting his own hair. Or the rendition of the Rikkai matches, which also include flashbacks of the Rikkai regulars anxiously waiting for Yukimura in the hospital, remarking on their True Companions bond.
For the Tenimyus, most of the Seigaku 2nd Cast (except for Shirota Yuu, Aiba Hiroki, Kujirai Kosuke and Araki Hirofumi, who were good singers and actors) the others were horribly out of tune. There were many instances where it was painful to hear them reach high notes (most notably all the songs that featured Adachi Osamu and Suzuki Hiroki singing).
It was justified with Yanagi Kotaroh (Ryoma), because his voice was affected by his car accident, but still no excuse for the others. Many fans still forgive his performance because of this.
Even thought Araki Hirofumi was a very decent singer (he even had a solo), fans still were fixated on the fact that he was very short to be playing Inui, since even Fuji (who is supposed to be the shortest of the seniors) was taller than him (also justified since Aiba Hiroki is taller).
The one who got more criticism was Adachi Osamu as Kikumaru, since he didn't look like the character at all, even with the wig on. Not helped by the fact that he wasn't a very good singer either.
For the Live-Action film, even if the film adaptation did or not an Age Lift, it was an abuse of Suspension of Disbelief the casting of RIKIYA as Egate Mcleod Higaki. You can seriously reassure the guy looks 15-19? That's also applied for the cameos of the rivals, many Tenimyu actors.