Modern Vespa

Stupid but how can you tell if you have a blown piston?

Oil - lots of it - in the airbox.
If you have a truly blown piston you have a lot less power. The incidents we've seen - like mine - the power is still ok, it just blows oil like crazy.

My MuZ 660 single is kind of like that if you overfill the oil reservoir under the back seat. It's a Yamaha dry sump motor and it uses some oil by design. You are told to check the oil level every 600 miles. I recently took it on a 3K mile trip. I found if I rode much over 4K rpm (70 mph) it lost oil faster, to the point it needed some more every 300 miles. If you stayed under 4k oil loss is half that like what is normal for this bike. I have yet to look @ the spark plug but will be doing that soon.

I talked to an owner of the same bike at a MC Rally I rode to and he said his MuZ is the same. There's no power loss but if you overfill the oil tank, smoke comes pouring out when you start it up like for mosquito abatement. Others think the motor is toast. Unfortunately it's easy to overfill this bike with oil.

I only ask because my 2009 MP3 500 is running like crap at low speeds. I am replacing belt, rollers and variator so hoping that is the issue.

Define "running like crap".
Details matter in what to look for.

When I am driving at slow speeds, starting from a stop it runs very crappy cuts in and out. I have replaced the spark plugs but wondered if I might need belt, rollers, etc.

How many miles on it?
Ever adjust your valves?
Otherwise, loose spark plug lead, bad plug, blocked fuel filter, broken connection on a fuel injector.

to add to Sully's list

when was the air filter cleaned and very lightly oiled. I stress lightly.
also have you ever cleaned the throttle body?
are you using premium gas?
when was the spark plugs last changed and what are you currently using?
when was the variation last serviced with new rollers?
did you clean the clutch?

Air filter was just cleaned and lightly oiled
I have not cleaned the throttle body
Yes I always use premium gas
Spark plugs were changed 2 weeks ago NGK CR7EKB I believe
I don't know any history on the bike as far as servicing.
17000 km 10,625 miles
Never adjusted the valves
Thanks for everyones help

With a blown piston you wont be able to start there will be no compression because there will be a hole or lrge crack in it

When the pistion rings are broken you can do this several ways

pull the headunit off and turn the crankshaft

or look with a endoscope iff the cylinder is pitted (small dents) small pieces of the rings will do that or scratch marks on de cylinderwall most likely you pistonrigs are gone

if you got only oil in the airbox please change out the reedvalve take the flywheelcover off and it sits above the axle off the waterpump blow on it if its not making any sound when you blow on it its gone

clean the throttle body and you are so close in mileage to the valve adjustment I would go ahead and do that as well. they are due every 12,000 miles.

Did this start with the sparkplug change or was it already running rough.
It would be helpful to put things in a timeline with symptoms perhaps.
In my experience a dirty/glazed clutch slips. Rough running mostly would point to spark and/or fuel. Did it sit for awhile? Any chance of water in the fuel?