I sold common stocks right at the end of 2016. The sale date was in 2016 and the settlement date was in 2017. In which tax year should I record the sales?

If the sale of the stock was "settled" in 2017 (settlement date shows as 2017), then the transaction would be reported in the 2017 tax year.  The settlement date is the date that the transaction is considered to have legally taken place.