What is the dance that Tsukihi does in the third opening of Nisemonogatari ("Platinum Disco")? It's the one where she does all of the hand movements.
It's based on bon odori, the dancing at O-bon festivals. O-bon is a holiday in Japan that takes place towards the end of summer. All the "summer festivals" in anime where people dress in yukata and go eat yakisoba and scoop goldfish and watch fireworks are O-bon festivals.
Here is a video that shows some people doing bon-odori. The video description offers some facts about the dancing:
Bon Odori (盆踊り), meaning simply Bon dance is a style of dancing performed during Obon. Originally a Nenbutsu folk dance to welcome the spirits of the dead, the style of celebration varies in many aspects from region to region. Each region has a local dance, as well as different music. The music can be songs specifically pertinent to the spiritual message of Obon, or local min'yo folk songs. Consequently, the Bon dance will look and sound different from region to region. Hokkaidō is known for a folk-song known as "Soran Bushi." The song "Tokyo Ondo" takes its namesake from the capital of Japan. "Gujo Odori" in Gujō, Gifu prefecture is famous for all night dancing. "Goshu Ondo" is a folk song from Shiga prefecture. Residents of the Kansai area will recognize the famous "Kawachi ondo." Tokushima in Shikoku is very famous for its "Awa Odori," or "fool's dance," and in the far south, one can hear the "Ohara Bushi" of Kagoshima.
Of course, Tsukihi is well known as a yukata freak who joined the tea ceremony club just to have an excuse to wear a yukata at school, and bon odori is often done in yukata. (You can even kind of see how the movements of the dance, which mostly involve the hands and arms, are adapted to the leg-restricting nature of the garment.) So that's why her opening features her doing bon odori.