i think sunday is the best as most people camp on the weekends and are leaving on that day so your chances are pretty good at snagging a nice spot with out a reservation.

 I don’t know if there is any type of survey on this.  My best guess would be a Tuesday or Wednesday.  I know that many people will go to the campground reservationless at 3-4am on a Thursday and sit outside the park office (lined up) and wait for the announcements of what is open. Then they book for 4-5 days.  Hope you get one!