Photo: CSI decorated with flowers in honor of שבועות - credit: CSI
קדיש - ליל ראשון של שבועות - S&P - London [EA]
In the Western Ashkenazi tradition the Chazon can freely choose to incorporate any of the (Western European) אקדמות melodies into various parts of the liturgy:
קדיש קודם ברכו/קודם מוסף - ניגון אקדמות - KAYJ [MF]
ברכות ק"ש - תתברך - KAYJ [MF]
ברכות ק"ש - וכולם מקבלים - KAYJ [MF]
קדיש תתקבל - ניגון אקדמות - KAYJ [MF]
On the first day, a נטילת רשות is chanted as an opener to קריאת התורה:
אקדמות - Western Ashkenaz [HB]
אקדמות - Western Ashkenaz [ADC]
אקדמות - Eastern Ashkenaz [EW]
A shiur from R' Eli Pick on אקדמות:
A shiur from R' Dovid Sulzbacher on אקדמות:
While the עשרת הדברות are being read, there is a custom for the congregation to stand:
Standing for Aseres Hadibros [R' Dovid Sulzbacher]
There is lengthy poem written by R Solomon ibn Gabirol entitled אזהרות which is chanted in parts over the festival. For more on that, see here [JC].
On the second day, a נטילת רשות to say תרגום is chanted as an opener to the הפטורה. See my article here
Shiurim from R' Eli Pick on עניני שבועות:
Were We Forced to Accept the Torah?
After שבועות, some congregations don't say תחנון:
Tachanun after Shovuos [R' Dovid Sulzbacher]