What color? If it's blue smoke, that is engine oil and is typical of worn valve guide seals. Oil runs down between the valve stems and guides after the engine is stopped and it's still thinned out from being warm. That oil collects on the valves and on top of the pistons, then burns off when the engine is restarted cold. That isn't particularly harmful but it can be irritating.
This also often happens after you've had a running problem that causes a lot of extra unburned fuel to go through the engine. That gas can dilute the oil, making it thinner and more likely to run down those valve guides. The problem may clear up shortly after the next oil change. If you've had such a running problem it's important to check the oil level. If it has gone up, that's because raw gas has run down past the piston rings into the oil. It is real important to change the oil and get that gas out of there. A little fuel will burn off as the engine runs but too much will destroy the oil's lubricating properties and cause severe engine damage.
White smoke is a sign of engine coolant. If that smoking only occurs at start-up, coolant could be seeping into a cylinder through a leaking head gasket while there's still pressure in the cooling system. Typically you'll see bubbles in the coolant reservoir too and the fluid level will go down over a short time.
Two tests can identify a leaking head gasket. One is to add a small bottle of dye to the coolant, then search a few days later with a black light, especially at the tail pipe. The dye will show up as a bright yellow stain. Your mechanic can also perform a "sniffer" test. That involves drawing air from the radiator through a glass cylinder with two chambers partially filled with a special dark blue liquid. That liquid will turn bright yellow if combustion gases are present. If there is, it stands to reason coolant will go out through the same leak, either while the engine is running or after it is shut off.