I found my stolen bike 4 days after it was taken in the thief's SUV, which was parked on a main road about a half mile away. Without police help, I narrowed down the daily patterns of the thief, his truck, his girlfriend, and ultimately, my bike (although, tbh, I thought I was screwed the entire time I was "hunting"). Some observations:
1. Moped thieves are opportunistic, and probably haven't thought the crime through too thoroughly (I mean, if I were going to commit a felony, I'd want to stand to gain more than a few hundred bucks). This is NOT a perfect crime; slip-ups are almost guaranteed. Ask anyone and any business nearby for surveillance footage. Maybe someone noticed a truck or van circling; go dig up evidence. Do this NOW before people forget details.
2. If this was taken from your home, the perpetrator probably lives nearby (i.e. you probably weren't cased, someone likely just noticed the bike while walking by for another reason). Who would normally walk within line of sight of the bike? Where would they be going?
3. Again, these crimes are almost never well planned. A buyer isn't lined up. This means that A) the thief probably still has possession of the bike, and B) he/she is hiding it because he/she doesn't quite know what to do with it. MAKE SURE EVERYONE KNOWS IT'S HOT. Put up posters, Craigslist ads, etc. This may force the thief to hold onto it (which hopefully keeps it close, and gives more time to slip up).
4. Finally, it doesn't run. Think about that - the thief likely doesn't know why it won't work; unless he/she is a 2-stroke pro, someone else will need to explain it to them. Mention this to mechanics, Autozone shops, gas station shops, etc. See if anyone's been asking strange questions about 2 strokes or "scooters."