remjax: The 770 (or some of the units) had a hardware screen problem. The N800 doesn't. However, there are some (a few) reports about another type of screen problem, which isn't the actual LCD but instead a sensitivity problem with the digitizer (touch-screen) near the side of the display. Others will have a better idea of how many units we're talking about, e.g. Texrat. I've only seen my own unit, and the hardware is good in every way.

Edit: About random reboots: Yes, I've experienced them. But, they _only_ happen when the unit is idle.. I've never had any type of crash of the OS while using the unit. I use it a lot. But, occasionally when it's just sitting there, not connected to a network but possibly searching, I suddenly hear the Nokia jingle and it has rebooted. The cause of this is known by Nokia (it's a watchguard program that detects that the DSP doesn't respond, and decides to deliberately reboot the unit in a controlled fashion -- this is not a 'crash and burn' thing). Presumably they have a fix too, in the near future.
But, as it only happens when it's idle.. it's never impacted my use of the N800. Then there's also a case of timed reboots around midnight or something like that, this is caused by the alarm system having somehow gotten an entry which tells it
to do so (a calendar event, if you want). I haven't seen this. In any case it's easy to fix.