So many of them!
Every culture around the world has superstitions for ringing in the new year. (Including the Chinese, although their calendar doesn't roll over until the lunar new year, which usually happens in late January or early February.) What all of these superstitions have in common is a belief that the things you do (or avoid doing) on the first day of the year set the pattern for all of the days that follow.
This is an understandable thing, given that the change of the calendar is a notable event every year. But the sheer volume of related superstitions is astounding. It rivals, in both number and variety, the superstitions surrounding major events like death or childbirth. Although maybe that makes sense, since the new year is when the old year dies and the new year is born.
The superstition which is the most commonly practiced in the United States has to do with eating beans (which symbolize both coins and plenitude) and greens (which symbolize money). Every year, grocery stores across America are stripped bare of their black-eyed peas, which are often cooked up with greens into a traditional Southern dish called "Hoppin' John."
(Although in my personal experience, the only thing Hoppin' John brings you in the new year is farts, and plenty of them.)
Aside from the traditional superstitions (clean your house to let the luck in; eat grapes to ensure prosperity) many people go through a more elaborate ritual known as "new year's resolutions." Although these traditionally prove no more effective a means for change than eating a plate of beans and sautéed greens, that doesn't stop us from making out a well-intentioned list every year, only to abandon it by the middle of January. But how else would the diet, exercise, and smoking cessation industries survive?
Of course, no discussion of New Year's Eve traditions and superstitions would be complete without a mention of Hollywood's belief that the person you kiss at midnight is your true forever beloved. What better time to find the love of your life (whether in a "meet cute" or not) than at a loud drunken party in the middle of the night, which was probably a work day for you as well? As unrealistic and downright bizarre as this belief may be, it still remains one of Hollywood's most sacred and cherished cinematic plot devices.
Honestly for most people, the most important new year's ritual to observe is the annual Drinking Lots Of Water And Taking Ibuprofen Before You Go To Bed.