Eileen has a crush on Rigby, as revealed in "Do Me a Solid". In recent episodes, however, Rigby has hinted that he might like Eileen back. As of now, they share a very close relationship and they deeply respect each other.
In the episode "Skunked", after he angrily grabbed her by the shoulders, held her in the air and shouted at her for laughing at his 'life' against his will, he was mortified at what he did and personally apologized to her, and ran off to find the Were-Skunk as the last straw. Rigby constantly tries to avoid Eileen, but tends to not succeed in his efforts to keep away from her.
Also, in "Camping Can Be Cool", Rigby seems to express friendship when he complimented her for breaking Margaret's car window in order to get in, and her ability to name constellations without looking when they were playing video games.In the episode "Diary", Rigby admits he thinks Eileen is hot without her glasses on.
In "One Pull Up", Eileen and Rigby spend time together to help Rigby do a pull up to keep his job. At the end of the episode, they hug, and even though Rigby cracks her spine, Eileen whispers that it's "totally worth it".
In "Tants", Rigby defends Eileen's work on the new tants. In response, Eileen gives him a smile, and he smiles back.
In "Return of Mordecai and the Rigbys", Eileen gives an overheated Rigby a towel. As he sensually wipes his face and moans, Eileen shows her inner fangirl and asks Rigby in a deep voice if he felt better. Eileen also has a conversation with Rigby and motivates him to not give up after the band splits.
In "Video 101", Rigby, along with Mordecai and CJ, help out Eileen with her video project. This is one of the few instances, along with "Tants" and "Take the Cake", where Eileen becomes visibly annoyed with Rigby.
In "Play Date", Rigby is shown to be at Eileen's house watching movies with her. When Mordecai asks why, Rigby replies "Very important!" in a defensive tone.
In "Skips in the Saddle", Rigby mentions that he is going over to Eileen's house to see her new rock polisher. When Muscle Man and High Five Ghost start teasing him, he defends himself by saying that him and Eileen are only hanging out as friends. When Mordecai later asks Rigby if he put his arm around her rock polisher, Rigby replies with his signature catchphrase, "Stop talking!"
In "Real Date", Eileen decided to give the dessert to Rigby, which made Rigby excited. After eating it, Rigby replied to Eileen and said "I love it!" And Eileen said "I thought that you would." They smiled at each other.
In Eileen Flat Screen, it is revealed that the duo actually hangs out alot when Mordecai is busy. They even went to a fair together offscreen. In the episode, Rigby displays what even Mordecai notes is an unusual degree of responsibility and conscientiousness for him as he sets up Eileen's new flat-screen TV for her, and sets up the TV and pizza delivery as an elaborate surprise for Eileen (which makes Eileen blush when she finds out Rigby was behind it). Overall, the episode revealed that their relationship is growing, and that Rigby may like Eileen more than just a friend.
In Merry Christmas Mordecai, Rigby and Eileen were talking and Mordecai said, "You know there's mistletoe between you," with Rigby saying "Yeah, so?" Mordecai then says, "You know that people have to kiss under the mistletoe, right?" An annoyed Rigby replies with "Yeahhhhh, but it's not like its a rule rule." Mordecai mocks Rigby by clucking like a chicken, suggesting that he thinks Rigby is being a chicken. Rigby then kisses Eileen on the cheek causing her to blush. Ever since they started hanging out most of the time, they both seem more comfortable being around each other, so Rigby might have a crush on her also.
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Regular show: Rigby Edition
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