When booting try holding down the four chord Command-Option-P-R and keep it held down through at least three reboot cycles, there will be a chime for each one; that often helps high-level hardware issues.
Next, boot and hold down the left Shift key as it comes up. This will bring up the system in Safe mode. Take note, this can be very slow, as the cache isn't used; this helps when something in your cache has been corrupted.
When using Disk Utility, make sure that you check both the disk (which does the structure and partitions) as well as the volumes under it (where the data is stored); do permissions as well. This should take a few minutes.
For diagnostic information, hold down Command-V (verbose), and a Unix-like log will scroll by. If it freezes, the last few lines should be useful in diagnosing.
Wanted to make sure that I understood you correctly, that you're seeing just a white screen at boot -- if so, the above should help glean some information. If you're going along and the screen turns white with a multi-lingual error message, that's the "white screen of death" and appears when there's a hardware problem. This can be an obnoxious USB device that isn't behaving, the wrong kind of memory was installed, failure on the motherboard, etc.
Most of the Mac problems I'm aware of have been when someone tries to save a few bucks in upgrading their memory. It's a complicated subject, but when RAM has to internally be refreshed on a clock cycle, it matters which part of the clock cycle that happens on. It appears that PCs and MACs do things slightly differently, and from what I've seen most marketing information the two products look identical. Carefully scrutinizing the hardware of the two products shows there is a difference.
I agree this sounds like a RAM issue. You might want to swap in some memory from a friend's machine and see if the problem goes away. Optionally, the recommended course of action is to use just the memory Apple supplied, see if this solves the problem, and if so, go to Crucial.com and use their tools to see exactly what's needed.