[hjqqmybr] 960 openings

To add an element of chance (for the purposes of lessening the importance of prepared lines), one can go the checkers route instead of the more drastic Chess 960 route: select openings -- positions generally considered equal relative to the initial position -- randomly.

How can this be done?  We wish not to exclude any possible chess game, so one of the possible openings is the initial position.  Then, some specialization on the most common lines?  Find 959 more such lines.  (Though perhaps 1024 total would be better with 10 coin flips.)