I have the random restart issue.
I have a 2.0 Mac Mini I Rented in November of last year with upgraded ram and HDD. I am using a LG 32LX2D-AA 80cm LCD. I purchased it with 10.4 and recieved the 10.5 drop in pack a couple of weeks later. I noticed no probs with 10.4 but I can't say for sure if I just missed it. Installed the drop in and only discovered the problem after noticing half my iTunes library had actually copied across from my old drive, this was due to a random restart that occurred while i was copying it. I successfully managed to witness it occurring with all external devices disconnected (except the LCD TV).
I have no scheduled shutdowns, it is not regular, it is random and not always a restart...
I contacted the store where I purchased and they suggested re-installing 10.5 to rule out if it was leopard or not so I did. At first it was ok but then instead of restarting it would go to "sleep" but was unable to be woken without holding the power button down. Eventually It began with the random restarts again. Sometimes it would restart a couple of times... the chime would sound, black screen, chime again, then when restart was almost complete, frozen screen, it would have a small glitch of about 9, inch long vertical black lines staggered in lines of three or there would be pixellated colour glitch of green and pink across the whole screen. This was when it was really bad and I couldn't get it working, even with a rest to cool down.
I have taken it in to the service department of the reseller three times now and every time over a week on the bench stressing the cpu and no fault. It's now been there for 30 days, 1.5wks on the bench no fault and apple have given me a case no# and a contact. All the time I am paying monthly rental for what was supposed to be a cool media centre.
I really believe that the mac mini is not compatible with LCD TV's and don't know what to do if that is the problem and a new replacement unit ends up the same, I have a 36 month contract of hire and can't afford now to get an apple display (which will be no where near as cool as my LCDTV)
I found these topics which I informed the service department of in email
http://discussions.apple.com/message.jspa?messageID=6914286#6914286
http://discussions.apple.com/thread.jspa?messageID=6914199#6914199 aswell as this one we are in and more which aren't necessarily so similiar.
Mac Mini 2.0gHz 2coreDuo Mac OS X (10.4.11) 2gb Ram (Factory ram installed on purchase by the reseller SwissBit i think)