REGA 3- Sounds better with dust cover ON of OFF ?

Don't know specifically about the Rega, but the general rule is -- dustcover off. Best answer -- listen for yourself and decide.
--Neil

I expierimented with having the cover on and off on my Rega P25 and liked the results better with my cover off. It seemed to "open" things up. Probably all in my head...but try it...it is a cheap mod.

If you leave it just open it wouldn't do any trick compared to the closed one. For the audiable difference remove it with hinges together(I know it's pain to place it back when you need to keep it closed other times).

Agree with above, especially Marakanetz. Remove the dustcover completely from the table during play, for best sound. Replace when finished.