It is a kabbalistic custom, mentioned in the Zohar, to stay up all Shavuot night learning.
I have other obligations, such as making sure the baby is put to sleep and taken care of the next day, so I do not know that I will spend the entire night doing so - I will probably turn in about 4 PM at the latest - but here is what I intend on learning:
I think the most appropriate thing to stay up all Shavuot night learning would be Shadal (Shmuel Dovid Luzzatto)'s Vikuach Al Chochmat HaKabbalah, in which he attacks kabbalah, the authenticity of the Zohar, and the kabbalistic practice of staying up learning all night on Hoshana Rabba, among other things as a violation of vesamachta beChagecha.
I've included the first two pages of the sefer, which contains this first attack. Note he writes as if in a conversation with someone attacking kabbalah and casts himself as the defender, but that is just a rhetorical device. Click on each of these two and zoom to full screen to see. I am not going to scan the rest of the sefer right now because I have more important things to do - get your own copy :)
Update: The fact that the benei yisrael slept Shavuot night at matan Torah, though, is found in Pirkei deRabbi Eliezer perek 40:
ר' חנינא אומ' בחדש השלישי היום כפול בלילה וישנו ישראל עד שתי שעות ביום ששינת יום העצרת עריבה והלילה קצרה ויצא משה ובא למחנה ישראל והיה מעורר ישראל משינתם ואמ' להם עימדו משינתכם שהרי אלהיכם מבקש ליתן לכם את הא"רץ התורה כבר החתן מבקש להביא את הכלה להכניס לחופה כדי ליתן לכם את התורה
That this was a bad thing, and that we need to "fix" it, I don't see here. That seems to be the kabbalistic twist on this.