1) I would not object to us no longer characterizing you as not my girlfriend
Sheldon asks Amy to be his girlfriend
2) The Tiara!!!
Sheldon and Amy encountered their first problem in their boyfriend/girlfriend relationship. When Amy got upset after Sheldon showed no reaction to her article making the cover of a prestigious journal, he turned to Leonard and Penny for advice on how to be a better boyfriend.
3) Amy's Campaign
Amy has embarked on a campaign to increase Sheldon's feelings for her. Their "romantic dinner" was a unique and successful one, at least for Amy. However, my favorite part is when Sheldon confessed to Leonard that he was having affectionate feelings for her and kept thinking about her at inappropriate times. The way Sheldon explained how the random particles morphed into Amy's dandruff on her slightly humped shoulders is just sweet.
Shamy's Romantic Dinner
Sheldon engages in the tradition of whining about his woman
4) Sheldon holds Amy's hand
Perhaps the greatest leap forward in Sheldon's willingness to show his softer, more emotional side to Amy is when he voluntarily, without any prior prompt from Amy, held Amy's hand while Howard's Soyuz capsule was being launched into space.
Sheldon holds Amy's hand
5) Sheldon romantic speech to Amy
Of course Sheldon will express his feelings this way. I can imagine him feeling awkward, don't know how to express the feelings that are new to him, and opted for hiding behind the Spiderman and gave that touching speech. The strength of Jim Parsons as an actor makes his words believable.
Sheldon romantic speech
6) Sheldon taking care of Amy
Many people like this episode because of the hint of "kinkiness" but I like the fact that Sheldon did it because “I made a commitment in writing to comfort you in times like this. And additionally, you are my girlfriend and I care about your well-being.” Sheldon doing almost anything that is justified in the name of science or reason. In that case, she was sick enough that she needed to be bathed; she needed to have something rubbed on her bare chest. (by the way, the bloopers from this episode are really cute)
Sheldon taking care of Amy
Jim and Mayim Bloopers - TBBT Season 6
Initially the pair took on more of an academic, personal relationship than a romantic relationship, gradually, the barriers started to drop, as Sheldon overcame his touch aversion. And before Sheldon noticed, Amy has become a comfort to him. Sheldon needed to be held and consoled by Amy when he is very upset about Kripke
Sheldon needed a hug
8) Amy is Sheldon's emergency contact
Sheldon always give the most enthralling and tantalizing gifts. Even before he started to really show some effort in being romantic in later seasons, he already showed his potential in gifts giving in Season 6 Valentine's Day episode. Amy is now his emergency point of contact at the university. (Well, I thought, if I have a stroke or a kidney stone, who would I want to share that with?)
Amy and Sheldon's Valentine's Gifts
9) Sheldon reveals if he will ever sleep with Amy
This is one of my favorite scene (both story wise and acting wise). I was shocked and thrilled to hear Sheldon admits that he is quite fond of Amy, that he’s working on his lifelong aversion to the physical contact that comes naturally to others, and that potentially, somewhere down the line, he could see himself actually sleeping with Amy. Sheldon’s honest admission shows that he may not always realize when he’s doing/saying something wrong but he is working on improving himself. Amy and Sheldon both give just as much to their relationship, they just do it in different ways.
It's a possibility
10) Love Spell
This is a lovely episode which managed to address character dynamics in believable fashion while throwing us the usual geeky entertainment with style.
Sheldon and Amy got physical in the context of Dungeons and Dragons. What maybe is a small step to other people, is as big of a step to these characters. Sheldon and Amy's frank discussion of what their relationship is illustrated that progress beautifully. Amy understands Sheldon. Sometimes she does want more from him but she also understands why their relationship is what it currently is. Despite Sheldon's issues, Sheldon gets Amy. While their progress toward intimacy has been moving at an incredibly slow pace, progress has been made—which shows that Sheldon is trying and that he does truly care about her.