UPDATED 4/13/13 9AM EST
Like I did with Super Sentai and Power Rangers, there are some storylines that repeat that become tropes sort of. Kamen Rider has some recurring themes and topics.
Grey-Area Organizations
These are organizations that are not exactly the villains, or don't start out that way. Some facilitate the gear and henshin devices of the Kamen Riders and some offer explanations or are rooted in the villains. Smart Brain, Board, ZECT, Foundation X, Kougami Foundation and a couple others. Foundation X and Kougami Foundation are connected and carry through from Double to Fourze. I might be missing some.
Multiple Users
There are different people that are one Rider, either one gets killed off and they don it or there are multiple versions of one henshin unit (like the RiotTroopers) or someone is given the henshin unit for one time. Kamen Rider 555 has more than five users (Kiyotaka Nishida, Yuji Kiba, Saya Kimura, Kyōsuke Tokumoto, Takahisa Shindō, Rina Abe), about four were in Kamen Rider TheBee (Sou Yaguruma, Shun Kageyama, Arata Kagami, and Masato Mishima), and Three Ixas (Yuri Aso, Megumi Aso, Kengo Eritate and the main guy).
Hero is one of the Bad Guys
Not exactly one of the bad guys but they end up being half of what the villains are. Takumi Inui /Kamen Rider 555 turns out to be an Orphnoch, Wateru/Kamen Rider Kiva in Decade was half-Fangire, and I think there were other Kamen Riders that were like that.
Hero should have been dead
This is the majority of the Showa era were the original Kamen Riders were at the brink of death or killed and made superpowered by some weird space rock.
Happy Rider, Brooding Rider
Here are some I remember: Kamen Rider Ryuki (Happy) and Knight (Brooding); Kamen Rider Fourze (Happy) and Cosmos (Brooding); Kamen Rider Decade (Brooding) and Kamen Rider Agito (Happy) and I think there are a few more. I think that Agito and Kaixa and also Blade and Garren, not sure.
The Tragic Anti-Hero or bad Rider dies
Ryuki has this in spades (Zolda, Knight, Ohja, Gai, Raia, etc.), Kabuto (TheBee, PunchHopper), and some in the movies of Faiz and Hibiki. Also not only riders but villains that redeem themselves and then die or sacrifice themselves.
The Tragic Anti-Hero or bad Rider dies
Ryuki has this in spades (Zolda, Knight, Ohja, Gai, Raia, etc.), Kabuto (TheBee, PunchHopper), and some in the movies of Faiz and Hibiki. Also not only riders but villains that redeem themselves and then die or sacrifice themselves.
The Girl's Brother
A girl (non-rider/love interest/girl friday) is common but that is a trope but not a storyline. But one common storyline is the girl concerned about her brother like Yui Kanzaki in Ryuki who was concerned about her brother Shiro, who was responsible for the powers and Hina concerned about her brother who is possessed by Ankh. Related: http://henshingrid.blogspot.com/2013/04/power-rangers-this-storyline-looks.html | http://henshingrid.blogspot.com/2013/04/sentai-this-storyline-looks-familar.html
