Piaggio engines all have 5 filters, except the 250 SL which has 4. They are:-

Oil filter (except the 250 SL)
Oil strainer
Air Filter
Fuel Filter
Transmission case air filter

K&N (WinCE)

When changing the engine oil, always remove the gauze strainer and wipe off any dirt. Make sure when refitting you push it fully home or it will be crushed by the drain plug when you refit it as it sits into the hollow plug.
On some models the air filter elements are foam and can be washed and re-oiled several times. Replacement foam can be obtained from Malossi and others. The 250SL and 500 uses a pleated paper element and can be blown clean a couple of times but then needs replacing. K&N make a suitable replacement for the 500 as it is exactly the same as early VW 1.6 Golf and Polo models, part number 33-2092 and costs around 30 and will last a lifetime, just clean and re-oil. Automatic transmission fluid is ideal for this.
The fuel filter is external on all models but the 500 where it is submerged into the fuel tank as part of the pump assembly. Most owners need not worry about these filters.
The last one is the transmission air filter. On the front end of the transmission case is a circular gauze filter plate inside a duct, and on the Evolution transmission cases, there is a foam strip running along the length of the casing. Not strictly a filter but a soundproofing item. This foam gets clogged with dust and just needs a blow through or a brush down whenever the casing is removed. The same goes for the gauze filter.
All these filters should be changed or cleaned during normal routine servicing and do not require any further inspection between times.
On the 500 it is possible to improve the air flow through the standard air filter casing by either shortening or removing the rubber snorkels from the box. This does not affect the running mixture, just slightly improves throttle response as does fitting a K&N filter element.