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.
Anti-Climax Boss: Unlike most villains, Kintoki Sakata poses almost no challenge to Gintoki alone and their fight lasts for less than ten pages.
Asexual: About half the fans insist that Gintoki is asexual. However, he's been shown spying on women in the shower, in the pool and on the beach. He also has a big crush on Ana Ketsuno and tried to hit on her once, but she was already married by then. He was also quite excited at the prospect of sleeping with the legendary courtesan Suzuran before he learned how old she was. On the other hand he resists most sexual and romantic advances from women and is comfortable in his bachelorhood, which may have something to do with how the women around him are either violent or obsessive when it comes to him. He has also bent Hasegawa over while drunk, even if both of them would prefer to pretend it didn't happen.
Base Breaker: Tsukuyo is one in Japan due to being a relative newcomer to the regular cast. She's still popular enough to rank 10th and 11th on the character polls, but she is also hated by some fans who see her as either a Spotlight-Stealing Squad or a threat to rivalling ships involving Gintoki, sometimes even both. She doesn't really count as one in the English-speaking fandom though, as she mostly remains a fan favorite there.
Crack Pairing: A series with this many characters who regularly interact with each other is bound to be rife in pairings, but the Japanese fandom really likes to get creative by shipping Sakamoto and Takasugi (who very rarely show) with the likes of Kondo and Hijikata (whom they've barely interacted with), as well as with each other (they were old comrades but they've only interacted with each other in a very recent manga chapter); there were also doujin artists who published the Gintoki and Hasegawa pairing long before episode 239. Shinpachi has also been paired up with Sakamoto, Takasugi, and various unexpected Shinsengumi members for some reason or another. Other odd pairings include: Hijikata/Katsura and Elizabeth/Gintoki.
Kamui isn't even the main villain like Takasugi is and has only had two appearances in the actual story so far, yet despite this he managed to rank 7th in his first popularity poll and to the surprise of many later on jumped up to 3rd in his next poll.
Of the notable Shinsengumi members, Yamazaki is probably the least important and the most plain, yet despite or perhaps because of this he remains pretty popular, having managed to consistently rank in the top ten on the popularity polls.
Tsukuyo was one initially, having only appeared in the Yoshiwara Arc by the time the votes for the second popularity poll were being submitted and managed to rank 10th on said poll, making her the second most popular female character behind Kagura. However, her role has since expanded, to the point where she got her own arc which came out right before the results of the poll was released. She has also become something of a Base Breaker to the Japanese fans.
Evil Is Sexy: Takasugi, Bansai and Kamui for the female fans.
Fan-Preferred Couple: Gintama has so far no Official Couples, and the chances of there being any are slim, but this has not daunted the shippers in any way. Okita/Kagura, due to the Foe Yay and Belligerent Sexual Tension between them, are ardently shipped together. A close second would be Gintoki/Tsukuyo, especially after the Ship Teasy Red Spider Arc. Yaoi couples such as Gintoki/Hijikata and Hijikata/Okita are also hugely popular due to a large part of the fanbase consisting of Yaoi Fangirls. In fact Gintoki/Hijikata could probably be considered the most popular pairing in the fandom. The Yuri pairing would be Otae/Kyuubei.
Gintoki and Hijikata's rivalry throughout the series is often seen in this light, which has helped make them probably the most popular Gintama ship, particularly in Japan. The anime adding filler episodes with more Foe Yay probably also helped. The ship actually got lampshaded in Episode 92 with Gintoki holding a mock manga cover with them in a... erm, suggestive pose.
Kamui got just a little too excited at watching Gintoki fight Hosen. Cue Kamui's unhealthy interest in him. Umibozu even said Yoshiwara doesn't interest Kamui, he only became its ruler solely to prevent anyone from getting close to Gintoki, who he's staked his claim on as future prey.
Okita and Kagura have this in spades, so much so that whenever the two interact, it's almost always some kind of Belligerent Sexual Tension.
Nizo and Katsura, with Nizo cutting his hair off and caressing it against his face, talking about how it was soft like a woman's. If he meant to piss off Gintoki, it worked.
Kamui and Takasugi also get some in Odd or Even.
Freud Was Right: Isn't that the Neo Armstrong Cyclone Jet Armstrong Cannon?
Growing the Beard: While Gintama got fans in the beginning, it was the Benizakura Arc that got people hooked.
Another example with Sugita. In 2013, there was a manga chapter where Gintoki told stories involving Phoenix Ikki from Saint Seiya. When Ikki debuted in the spinoff Saint Seiya Omega the same year, he was voiced by none other than Sugita.
Idiot Plot: Arguably, anything that doesn't count as a serious arc has shades of this.
Memetic Mutation: The "strawberry milk speech" and "evil grin" pictures are recognizable even to people who have never heard of the series.
Nightmare Fuel: Sacchan and Okita's ideas on how to improve the series in Episode 50.
Pandemoniums also count when they're not seen by Shinpachi.
Periphery Demographic: The series is technically shonen, but thanks to its avoidance of most pitfalls of the shonen fighting genre (they don't even have named techniques, much less new ones every arc), deviation from "fight progressively stronger enemies" formula, and an older-than-average protagonist, it's attracted a lot of older fans who otherwise find JUMP comics juvenile. The presence of attractive guys who aren't entirely effeminate has earned it a healthy-sized Estrogen Brigade as well, in addition to a quite large cast of proactive and useful female characters who can fight (and be successful at it) and who aren't the usual flat and ineffectual Fanservice on legs..
Self-Fanservice: Despite a Running Gag of Shinpachi being entirely generic and plain-looking in the actual series, doujinshi artists consistently draw him incredibly pretty. It's surprisingly averted with Kondo and Hasegawa, though.
The Owee Arc displays very little knowledge of even the most well-known Wii gimmicks, and one of the launch titles is a dating sim involving groping, which doesn't really gel with Nintendo's famously kid-friendly image.
To be fair, the arc appeared was around the time of the launch of the Wii and it was making fun of the hype around getting one before lampooning video games in general (thus you have Tetris, a dating-sim and an RPG). It wasn't really meant to satirize Nintendo itself.
Unfortunate Implications: The romance in Gintama is pretty much, All Love Is Unrequited. The mass majority of couples don't happen because of intense dislike from one party or another. But in the case of Otea and Kyuubei the biggest reason for them to not get together is because they are both girl.
The Woobie: Hasegawa is the premier example of the series. He may be The Chew Toy, but it's hard not to feel sorry for him at some point.
Woolseyism: The official translated manga version pulled these out here and there. One particular instance involved a character shocked by how terrible a pun was. Rather than come up with an ill-fitting translation, they pointed out that the character made a Japanese pun, and was shocked that someone would do untranslatable humor—"That's the worst kind!"