Here's a minibike running on a wood gassifier:
You would need either a four-stroke or oil injected engine to do this, not to mention the big trailer full of burning stuff.
Also, most methanol is made from either coal or natural gas so it's not really a biofuel.
Ethanol or methanol might be good for a race bike since they're really high octane and (IIRC) burn a little cooler than gasoline, but I don't think either one would be practical for everyday use.
If there was such a thing as a diesel moped you could easily run one on biodiesel or maybe even straight vegetable oil.
Personally, I'm waiting until someone makes a steam-powered moped that I can run off of burning arugula.