The most distinctive Shavuot custom is undoubtedly the recital of the Azharot and the Book of Ruth that takes place before the Minha service. It is a shame that, due to the comparatively low attendance at this time of day, this beautiful poem and its distinctive S&P chant, are little known.
The main poem is a precis of the 613 commandments written by R. Shelomo ibn Gevirol (c. 1020-1058). In England the first section, summarising the positive commands, is read on the first day, and the second, summarizing the negative ones, on the second.
The poem is preceded by an introductory poem by R. David Eleazar ibn Bakuda, who lived in Saragossa in the 12th Century. He has followed the custom of hiding his name in the opening letters of the verses.
Both the Azharot and the Book of Ruth, are read one verse per person, among the men present.
The final section of the Azharot on each day is chanted to the tune of "Leshoni Bonanta" from the Prayer for Dew sung on Passover:
The Amsterdam and New York chants for the Azharot are totally different. Here's a clear recording of the New York version. I believe this chant is also used for Yigdal.
♫ NY version of the Azharot [Zachary Edinger]