Did you realize today is odd?
Next lineup of any sort is 05-06-07.
Next lineup of odds is 05-07-09.
Next lineup of primes is 05-07-11.
Next lineup of evens is 04-06-08.
Next triplicate is 07-07-07.
Next month-day-year palindrome is the whole month of July, 2007, from 07-01-07 to 07-30-07; 07-11-07 and 07-22-07 are especially palindromic dates within the month.
Next true mirror palindrome is 07-11-70.
Next set of squared dates is 01-04-09 (1 squared, 2 squared, 3 squared).
Next set of cubed dates is 01-08-27.