I'm just porting some code to the Due, which is fairly mature and stable, and has been running on AVR boards (all the standard flavors) and also on Maple5 (which is also ARM Cortex-M3).
The code seems stable on the Due except when the programming USB port is connected, in which case I'm seeing resets every few seconds (maybe get to about 6 or 8 seconds into the program execution and then a reset, and see all the setup() sequence again).
OTOH, if the program is left running without the programming USB port connected, it will run indefinitely without a problem, at least in my testing so far.
The code is fairly interrupt intensive, using one external interrupt. I'd estimate the sketch is spending somewhere between 10-20% of it's total time servicing interrupts.
Before I start digging further, does this ring any bells? No problems when running on any the AVRs or the Maple5 with the USB serial port connected, for example.