The whole anime was built on and driven by Homura's feelings for Madoka.
This couple was shipped even before the anime was aired because of their promotional art.
Followed by plenty of official arts of these two hugging, dancing, holding hands lying on grass, Madoka looking at Homura in locker room, playing piano and singing together, building sand castle at the beach.
At what point◊ does it officially stop being subtext? Because if we're not already there...
That's a good question, given that for Valentine's Day the mobile game released a short comic in which Homura gives Madoka chocolates claiming that she is her 'most precious person', and Madoka accepts them, saying that she also likes/loves (the word used is 'daisuki') Homura a lot. Hmm.
Homura's fixation on Madoka is infamous, not only in the anime but also portrayed in Oriko Magica manga, in Drama CD 1 'Memories of You', and Drama CD 2 'Sunny Day Life'.
She verbally admits to not giving a fuck about anyone who is not Kaname Madoka (Episode 7).
Since Homura keeps very close eyes on Madoka, always being in time to save her, it's no surprise that many think of her as a stalker.
Combined with her time stop power, Perverted Homura has become a popular fanon interpretation.
Homura threatens to kill Sayaka - and has real intent to do so - if the latter continues making Madoka sad.
Homura practically confessed to Madoka twice.
In Episode 8, she confronted Madoka right after saving the latter.
In Episode 10, the truth is laid out on why and to what extent Homura cares about Madoka. She became a Magical Girl in the first place for the chance to change history and keep her alive. It only gets slashier from there; possibly the best non-angsty moment (er, overly-narrow-superlative, sorry) is the way Homura blushes when Madoka glomps her.
The ribbons may be a shout out to Lyrical Nanoha, seeing as to how Nanoha and Fate are another infamously popular couple in anime fandom with waterfalls of subtext.
Remember Madoka telling Homura that she will always be with her? Fans have used this to invert the Stalker Homura meme, by having Madokami stalking Homura with her omnipotence.
The naked space hugging scene is, if possible, more blatant on the DVDs. How? They took out the sparkles.
The entire Homura x Madoka bond is cast more romantically in the manga. After she heard of the dream Madoka had, Sayaka's mental image of them casts them as lovers.
It's got even worse. Remember in episode 12, when Madoka finally left Homura after giving her her ribbon? In the manga, she takes Homura with her into godhood.
Even ancillary material most of the casual audience won't see is permeated with Les Yay.
In the Anicom magazine's 100 Questions, Madoka answers "Is there a boy you're thinking about?" by talking about how much mischief her little brother gets into. Then she describes her ideal husband as someone like her mother. In the PSP game, Junko mentions that Homura acts just like she did at that age.
Visitors to the Movic Madoka Magica exhibit got to hear Madoka gushing about how pretty Homura's hair is, and Kyubey telling Homura how Madoka thinks she'd look cute in glasses. Homura apparently goes red upon hearing this.
During the PSP game, Madoka gets so attached to Moemura it can actually make her uncomfortable. Even after Homura drops the glasses and starts distancing herself, she can't stop Madoka from taking her home to meet her parents, who are quick to mention she's all Madoka talks about recently. Fish, meet barrel.
Kyouko x Sayaka
The second pairing has a different flavor compared to the main couple.
The mobile game includes a New Year's special with the cast visiting a shrine and picking omikuji. Everyone's fortune looks good...except Sayaka's, which says "the person you're waiting for won't come". Kyoko, trying to cheer her up, promises that she'll stay with her.
The PSP game adds more. In both Sayaka's and Homura's route, it's possible to prevent Sayaka's transformation into Oktavia. In Sayaka's route, this involves Kyoko comforting her after they learn the truth about soul gems. In Homura's route, it requires taking Kyoko to talk with her after the freak out - Kyoko is the only one who can get through to her. Not her childhood best friend, not her sempai, Kyoko.
This pairing was the largest ship before Homura's story was revealed.
While Mami met and recruited both Madoka and Sayaka at the same time, she was obviously more taken to Madoka.
Shipping fully fueled by Madoka's You Are Not Alone speech to Mami, and the latter's euphoria when Madoka agreed to be by her side (Episode 3).
Mami x Kyouko
Mami x Kyouko has recently gained popularity as a result of 3rd Drama CD, Farewell Story, which revealed that Mami and Kyouko used to be quite close before the anime events.
In the Drama CD "Sunny Day Life", the only reason Kyoko comes to Mami's apartment is because she is serving cake. Oh Mami, seducing young inexperienced girls with cake.
Madoka x Sayaka
Not a popular pairing but some fans do ship them.
In Episode 1, Sayaka say to Madoka "I'll make you my bride!". Of course the scene is only a joke between friends.
Hitomi: You mean you can understand each other just by gazing into each other's eyes already?! Ah! I can't believe you went that far in just one day! What did you two even do last night?!
Sayaka: It's not really...
Madoka: Well, a lot happened, but...
Hitomi: Two girls can't do that! That's forbidden looove! (runs off)
Hitomi's playful act had become rather famous because of its humour and the irony that this show is full of implied "forbidden" romance.
During Mami's route in the PSP game, Sayaka gets so distracted by Madoka and Moemura's womance she starts ignoring her actual love interest.
Relevant information
Gen Urobuchi (series writer) said in an Madoka Magica interview (see Black Past) that he believes in same-sex relationships, and that they are not different than any man-woman relationship. He also said that if he wanted, he could write a BL story next.
In another interview, Gen revealed Madoka Magica was started and revolved around the concept "where Homura saves Madoka". Yuri is the pivotal idea of the franchise!
Ume Aoki (original character designer) and Hanokage (manga adaptation artist) ship Homu/Mado and Kyou/Saya.
on the Puella wiki (which links back to this page).
