Solution:
Oven Cleaner
Maybe it seems like an obvious solution, but it took almost a week of having a big ugly burnt-food spot on my brand new stove, and trying every cleanser and method of scrubbing and scraping, for me to discover it. Just spray onto the offensive spill, let sit for around an hour (depending on intensity of spill and burnage) and voila! It wipes right off. Works every time.