One of the most widely known superstitions that us Westerns get from the Scottish is known as "First Footing".
The 'first footer' is the first man to cross the threshold after the hour of midnight has struck: if h is dark-haired and carries with him such objects as coal, bread, salt and money, the good luck of the household is guaranteed for the whole year ahead. If, however, the first footer is blonde or red-headed, bad luck will befall the house; if the first footer is a female this is even worse, as she ushers in only the direst misfortune.

In the ideal world the first footer will
be a stranger to all present and on no account must he be cross-eyed, flatfooted or have eyebrows that meet in the middle. Once let into the house, he may then be shown out again by the back door, thus symbolically letting in the old year out.

Other superstitions include:

Whatever happens on New Year's Day sets the pattern for the rest of the year.

It is unlucky , therefore, to see the New Year in with no food or drink in the cupboards. They will remain bare for the
ensuring twelve months. The same is said
to be true with money.
 

Rise early to ensure a busy and profitable year at work, all those in employment should do something that reflects their work in some way, even if they take the day off.

Wearing something new on New Year's Day promotes the chances of receiving further new clothing in the year ahead.

Washing clothes on New Year's day is inadvisable, for one of the family will themselves be 'washed away' in the months ahead.

It is bad luck to pay money or loan money, lend precious belongings or break anything. Ideally, any outstanding debts should be paid by New Year's eve to ensure further debts are not incurred in the year to come.

It is also thought weather plans a role in the day. If the wind blows from north, bad weather is in store; if it comes from the south, fine weather and prosperous times lie ahead; if it blows from the east, famine or some other calamity is on the way; if it blows from the west, the year will witness plentiful supplies of fish and milk but will also see the death of a very distinguished personage.  If there is no wind at all, a joyful and prosperous year may be expected by all.

Babies born on New Year's day will grow up with luck always on their side.

Two other superstitions that come from my mother are:
 Place a silver dime above each door of your house at precisely 11:59pm and money will flow into the household all year.

Place a wrapped piece of cabbage into your wallet to ensure it will never be empty throughout the coming year.
 

source for some of this information from
Cassell "Dictionary of Superstitions"