Shizaya. Erika is a fan, if only to be refuted by everyone else's horror. There's also Mikado/Kida (Heterosexual Life-Partners if we remember the point that Kida was practically begging for Mikado to come and save stay with him), Shizuo/Tom (Tom is the only one who can keep Shizuo calm), Shizuo/Kasuka, Izaya/Kida (more Foe Yay, though to Izaya, Kida is merely a sacrificial piece that he thinks of as a little brother), Izaya/Mikado (Izaya mainly likes getting too close to the guy's space, though anything with Izaya usually develops into Foe Yay), Shizuo/Dotachin, etc... There's also Mairu, one of Izaya's younger twin sisters, who's willing to take a girl if she applies (since the spot for a guy in the sisters' heart is taken up by one Heiwajima Kasuka, in a Fangirl kind of way. Celty also tends to have girls who (apparently) believe her to be a man and crush over her.
In regards to Izaya/Kida: Kanra and Bakyura spend the majority of their time flaming each other on the chat boards, which Kanra jokes is because Bakyura's just totally Tsundere for her. Now add in the fact that Kanra is Izaya and Bakyura is Kida, and that they both know it...
To just touch on the Shizuo/Izaya subject with regards to adaptation differences. The ship is undeniably intensely popular, and often a flamewar topic in and of itself, but a lot of the popularity and serious fanworks of it come from the anime, and for not completely insensible reasons. In the anime, Izaya and Shizuo do have a hate-obsession relationship, with Izaya playing mind games with Shizuo, the clashes of intense violence, the hate-at-first-sight beginning etc. If you look at it in a certain way, yes, you can read a lot of subtext into their relationship, and with a little Alternate Character Interpretation you can make things even worse. However the same cannot be said in such a literal way for the light novel. In the light novel they don't go looking to fight each other as much, tending to deliberately stay out of each others' way, Izaya mainly tries to ignore Shizuo and kill him from afar without getting personally involved, and their relationship tends to look more like plain loathing. Both anime and novel are of course similar, and you can read the same relationship from both but you often find people arguing on the differences, and this is one of those cases where the adaptation changes the relationship dynamic enough to make it... interpretable in different ways. This should be kept in mind by anyone wanting to go out and participate in any Shipping wars!!
Ho Yay between Shizuo and Tom, while not prominent in the anime, gets increasingly apparent the more their relationship gets focused on in the light novels. Tom, for example, is the only person allowed to ask about Shizuo's sex life without getting punched in the face (and the first person to ever tell Shizuo that he's pretty).
Mikado and Aoba. As soon as Aoba is introduced in the novel, he instantly latches onto Mikado. Then you realize he's the leader of Blue Squares and simply wants to control Mikado. When it seems like he'll get his wish, Mikado retaliates by stabbing him with a ballpoint pen (thus earning his title as Memetic Badass and telling him that he stabbed him for involving Anri in his little game. Now the two are working together in the merged Dollars/Blue Squares group and Aoba seems to have taken Masomi's place in Mikado's life...almost completely.
Izaya admitting to Aoba that he's a masochist who is still looking for a special someone to be his sadist. He also states that he wouldn't mind being used by a better manipulator than himself and that he would admire that person due to their superior skill. Said person very well could be Jinnai Yodogiri, the main antagonist and ultimate manipulator of the whole Durarara!! series, whether or not Izaya had him in mind at the time of the converstation.
There is a bit between Anri and Mika, and Anri's feelings towards Celty are very crush-like
Speaking of which, in the manga, after Celty saves Anri from Niekawa's father, who was possessed by Saika, she blushes and invites Celty into her house for tea, which was absent from the anime.
Izaya and Shinra seems to have something going on in middle school. I mean Shinra took a knife for Izaya and Izaya's reaction is to extract revenge on the person that stabbed him, Nakura, and continue to to torment him well after they graduate. This is one of the few times that Izaya shows human behavior.
Now we also have Mairu and Kururi, Izaya's younger twin sisters, who randomly share an intimate kiss after beating up some thugs in DVD-only episode 25.