White smoke that quickly disappears is water vapor. This is sometimes caused by condensation inside the silencer.
White smoke that doesn't evaporate is often oil smoke.
This can be caused by a leaking valve seal or in the worst case, a worn piston ring.
Smoke from worn piston rings doesn't go away after the engine is started and run for a while.
Smoke from worn valve seals usually disappears shortly after starting after the small amount of oil that has leaked into the inlet or exhaust port is washed/burned away.
Oil burning caused by worn valve seals can be lived with for a long time without causing any major problem although if it is severe, it can foul the spark plug.
There are a few products on the market that can be added to the oil to soften the valve seals which can reduce/stop the oil leakage however they can also soften the other seals in the engine which can reduce their resistance to wear.
Oil burning caused by worn piston rings will worsen to the point of laying down a smoke screen as additional hours are run. Usually the law (and environmentalists) takes a dim view of smoke screens.
edit:
I also forgot to mention that these older engines can "wet sump" which is another way of saying, collect excess oil in the crankcase.
These engines are "dry sump" engines which rely on a oil pump to pump the oil out of the crankcase as it is running. (The oil is stored in a special oil sump that is not directly connected to the crankcase where the flywheels and connecting rod run.)
If oil leaks into the crankcase while the engine is shut down it will be thrown up onto the cylinder wall in such large amounts that the oil wiper ring and piston rings can't keep it out of the combustion chamber. If this happens, when the engine is started it will smoke rather badly until the oil pump can reduce the volume of oil in the crankcase.
To minimize this it is suggested that the piston be raised by using the kick starter to Top Dead Center when the engine is stopped so the leaking oil can't drain out of the crankshaft into the crankcase.