Generally speaking, you can file a tax return as soon as you have all the relevant documents needed to prepare your tax return. This might include such documents as Form W-2 reporting wages and tax withholding, Form 1099 reporting bank interest, unemployment benefits, and other types of income, and so forth.
Some people who's only income is from freelancing or farming might be able to file their returns sooner, as long as you have tallied up your income and related expenses for the previous year.
One word of caution: try not to file a tax return using the information from your last paystub of the year. There's three reasons why. First, your last paystub might not reflect exactly what you earned for the entire year. Second, your paystub does not include all the relevant information needed for the IRS to process your tax return (for example, the employer identification number of the business). Thirdly, the IRS is on the look out for people who file only using their paystub.
In a reminder notice I received in the mail just today, the IRS is instructing tax preparers not to file returns using a paystub. Rather, the IRS prefers for employees to first wait for their W-2, and if a W-2 has not arrived by February 14th, to call the IRS. The rationale here is pretty straightforward: the government wants to know if a company hasn't sent out their W-2 Forms, as this can be an early warning signal for compliance problems that the IRS could then investigate.