Narrowing Down Source of Instability

if it's blue screening, can you try to get a picture of it, or write down what it says... different error codes mean different things, so that would be a good indicator of whats happening

u can use bluescreen viewer to see whats causing the errors. if your machine is logging them.
if not make sure high precision event timer is enabled in bios set it to match your o.s bit rate (32 for 32bit 64 for 64bit) and that the rest of the bios is running optimized default with peg mode enabled for the gfx card.

Centaurd: Sorry for the delay. I really don't know if it's a motherboard problem. I wish I knew what the problem is, because it's still happening to me regularly enough to cramp my style in a big way.

nna1 and HEXiT: I have crash dump files for some, but not nearly all, of the crashes. The most recent one is from today (9/4/2012). The odd thing about today's crash dump is that the machine crashed multiple times today, but there was no BSOD on any of them: one screen-freeze/unresponsive at 12:09 PM (required switching off my PSU), one horizontal distortion between 5:00 and 6:00 PM, and another horiz. dist. a little after 9:00 PM. Usually I get no dump file without a BSOD. The computer was freshly started up on the first two crashes, nothing but a conservative group of startup programs running. The third time, I had the startup programs plus a few tabs in Firefox running. I've been running a whole lot more than that for over an hour now with no crashes.

The crash dump below is, I believe, from the second crash. The computer sat unresponsive with the horizontal distortion for possibly up to an hour before I hard-restarted it (my only option). WhoCrashed reports it as 10:08:34 PM GMT / 5:08:34 PM my time. Windows says the matching file was last modified at 9:22 PM, which is probably about the time I turned it back on after the second crash and right before the third.

On Tue 9/4/2012 10:08:34 PM GMT your computer crashed
crash dump file: C:\Windows\Minidump\090412-34304-01.dmp
This was probably caused by the following module: ntoskrnl.exe (nt+0x70040)
Bugcheck code: 0xA (0x24F, 0x2, 0x1, 0xFFFFF800030DD183)
Error: IRQL_NOT_LESS_OR_EQUAL
file path: C:\Windows\system32\ntoskrnl.exe
product: Microsoft® Windows® Operating System
company: Microsoft Corporation
description: NT Kernel & System
Bug check description: This indicates that Microsoft Windows or a kernel-mode driver accessed paged memory at DISPATCH_LEVEL or above.
This appears to be a typical software driver bug and is not likely to be caused by a hardware problem.
The crash took place in the Windows kernel. Possibly this problem is caused by another driver which cannot be identified at this time.

I can provide a total of 12 crash dump files if you'd like: 8 refer to ntoskrnl.exe, 2 refer to win32k.sys, 1 refers to ntfs.sys, and 1 refers to atikmpag.sys. (The last one is almost 2 years old and may be from when I was first setting up my two monitors; it predates the major crashing problems I've been having.)

Jimulacrum - I am having this exact same issue w/ my HP p6654y that I've recently bought secondhand. Originally I bought a new PSU, new video card, and new hard drive. I had a buddy look at the machine and he pulled a stick of RAM, which did the job and the computer worked fine the rest of the day. This left me to believe it must be bad RAM slots on the motherboard, but then the next morning the issues started again. I REALLY need to know if you ever came across a permanent fix for this machine!