Update:
I've got it working after cutting the 12V trace on my shield to the Vin pin on the Arduino and connecting a 5V supply to the 5V pin instead.
I suppose this means there is something wrong with the Arduino's voltage regulation circuit, or there is something that I'm not aware of.
On a sidenote, I've also noticed that the Vreg on the R2 runs cooler than the one on the R3, probably due to the package size.