You file one tax return for the entire year. Most people are on a calendar year (Jan 1 through Dec 31). Changing to a different year is difficult and probably requires an accountant to help, and is not recommended for individuals (sometimes companies do this).
You can't file a return for half the year before the divorce and half the year after the divorce. One return for the whole year, containing all the income, deductions and credits for that year.
If you are single on December 31, you file as single (or maybe as head of household) even if you were married for part of the year. If you remarry to spouse #2 on or before December 31, then you file as though you were married to spouse #2 for the entire year, even though you were married to spouse #1 for part of the year and single for part of the year.
