You file your taxes for the whole year using your marital status on the last day of the year. So you will file as single (unless one or both of you has a qualifying child dependent that allows you to file as head of household.)
Unless your divorce agreement specifies otherwise you would be considered to each own half the house and be responsible for half the capital gains taxes, if any. You will each also use half the adjusted cost basis (original purchase price plus certain closing costs plus improvements). In turbotax, each former spouse would report they sold a home they owned and lived in, to qualify for the capital gains exclusion of $250,000 per person. So you would report a cost basis of $490K and a selling price of $1M for your half-share, and your spouse would report the same. You have a capital gain of about $500,000, of which you can exclude $250,000 and pay capital gains tax on the other $250,000.
Allowable adjustments to the cost basis are listed in publication 523 beginning on page 8.
https://www.irs.gov/pub/irs-pdf/p523.pdf
If the divorce order specified something different (like a 60/40 split), you would report the selling price and the adjusted basis using that ratio instead.
